AUTOMORPHISM GROUPS OF GRASSMANN CODES

SUDHIR R. GHORPADE AND KRISHNA V. KAIPA

Abstract. We use a theorem of Chow (1949) on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of Beelen et al. (2010) concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chow’s theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alternative short proof of MacWilliams theorem concerning the equivalence of linear codes and a characterization of maximal linear subspaces of Schubert divisors in Grassmannians.

1. Introduction

Let C be an [n, k]_{q}-linear code, i.e., C be a k-dimensional subspace of the ndimensional vector space 𝔽_{q}^{n} over the finite field 𝔽_{q} with q elements, where q is a prime power. Automorphisms of C are basically transformations of the ambient space 𝔽_{q}^{n} that preserve the code C and the coding-theoretic properties of C. These come in different flavors: permutation automorphisms, monomial automorphisms, and semilinear automorphisms, thus giving rise to groups PAut(C), MAut(C), and ΓAut(C) respectively. The last of these, being the most general, will simply be referred to as automorphisms and we will just write Aut(C) instead of ΓAut(C). Let us recall that PAut(C) consists of permutations σ ∈ S_{n} such that (c_{σ(1)}, …, c_{σ(n)}) ∈ C for all c = (c_{1}, …, c_{n}) ∈ C. Equivalently, PAut(C) consists of permutation matrices P ∈ GL(n, 𝔽_{q}) such that cP ∈ C for all c ∈ C. Likewise, MAut(C) consists of monomial matrices M ∈ GL(n, 𝔽_{q}) (i.e., matrices of the form P D, where P is a permutation matrix and D a diagonal matrix in GL(n, 𝔽_{q})) such that cM ∈ C for all c ∈ C. Finally, Aut(C) consists of compositions Mμ of (linear maps corresponding to) monomial matrices M ∈ GL(n, 𝔽_{q}) and field automorphisms μ of 𝔽_{q} (giving maps of 𝔽_{q}^{n} into itself by acting on each of the coordinates) such that cMμ ∈ C for all c ∈ C. The group Aut(C) of automorphisms of C can be viewed as a subgroup of the group ΓL(n, 𝔽_{q}) of semilinear transformations of 𝔽_{q}^{n}. A classical theorem of MacWilliams [14] implies that Aut(C) (resp: MAut(C)) is the same as the group of semilinear (resp: linear) isometries of C, where by an isometry of C we mean a bijection of C onto itself that preserve the Hamming metric.

2010 Mathematics Subject Classification. 14M15, 20B25, 94B05, 94B27.

Key words and phrases. Grassmann variety, Schubert divisor, linear code, automorphism group, Grassmann code, affine Grassmann code.

Both the authors are partially supported by the Indo-Russian project INT/RFBR/P-114 from the Department of Science & Technology, Govt. of India. The first named author is also partially supported by an IRCC Award grant 12IRAWD009 from IIT Bombay.

Determination of (either or each of) the automorphism group of a code is of considerable importance. Besides being of interest in itself, this can be useful in decoding and often finds applications to the theory of finite groups. Complete determination of the automorphism group has been achieved thus far in a few select classes of linear codes, such as Hamming codes, Reed-Muller codes, etc., and we refer to the Handbook article of Huffman [18, Ch. 17] for more details. See also the more recent work of Berger [4] where the case of projective Reed-Muller codes is studied. We consider in this paper the problem of complete determination of the automorphism group(s) of Grassmann codes and also related linear codes such as affine Grassmann codes and Schubert codes. Grassmann codes were introduced by Ryan (1987) in the binary case and by Nogin (1996) in the q-ary case, and have been of considerable recent interest (see, e.g., [16, 7, 8] and the references therein). However, as far as we know, automorphism groups of Grassmann codes and more generally, of Schubert codes have not been studied so far. Affine Grassmann codes were recently introduced and studied by Beelen, Ghorpade and Høholdt [2, 3]. In [2], it was shown that the permutation automorphism group of the affine Grassmann code C^{𝔸}(ℓ, m), corresponding to positive integers ℓ, m with ℓ < m, contains a subgroup isomorphic to a semidirect product M_{ℓ×ℓ′}(𝔽_{q}) ⋊ GL_{ℓ′}(𝔽_{q}) of the additive group of ℓ × ℓ′ matrices over 𝔽_{q} with the multiplicative group of ℓ′ × ℓ′ nonsingular matrices over 𝔽_{q}, where ℓ′ = m − ℓ. In [3], this result is extended to show that PAut(C^{𝔸}(ℓ, m)) contains a larger group, say ℌ(ℓ, m), that is essentially obtained by taking the product of the general linear group GL_{ℓ}(𝔽_{q}) with the semidirect product M_{ℓ×ℓ′}(𝔽_{q}) ⋊ GL_{ℓ′}(𝔽_{q}). It was also shown in [3] that PAut(C^{𝔸}(ℓ, m)) can, in fact, be larger when ℓ′ = ℓ, i.e., when m = 2ℓ, and it was remarked that the complete determination of PAut(C^{𝔸}(ℓ, m)) is an open question. In this paper, we settle this question and show that PAut(C^{𝔸}(ℓ, m)) coincides with ℌ(ℓ, m) when m ≠ 2ℓ, whereas if m = 2ℓ, then PAut(C^{𝔸}(ℓ, m)) is a certain semidirect product of ℌ(ℓ, m) with ℤ/2 ℤ. Moreover, we completely determine MAut(C^{𝔸}(ℓ, m)) as well as Aut(C^{𝔸}(ℓ, m)). This is, in fact, facilitated by going back to the origins of affine Grassmann codes, namely, the Grassmann codes, and the corresponding projective systems, viz., (the 𝔽_{q}-rational points of) Grassmann varieties with their canonical Plücker embedding. Thus, we first tackle the problem of determining the automorphism group of the Grassmann code C(ℓ, m) corresponding to the Grassmannian G(ℓ, m) of ℓ-dimensional subspaces of 𝔽_{q}^{m}. Here it is actually more natural to deal with the corresponding projective system P = G(ℓ, m) viewed as a collection of n = #G(ℓ, m)(𝔽_{q}) points in the Plücker projective space ℙ^{k−1} = ℙ(∧^{ℓ}𝔽_{q}^{m}), where k = \binom{m}{ℓ}. Before proceeding further with the description of our results on automorphisms of Grassmann codes and Schubert codes, it seems worthwhile to digress to discuss the notions of projective systems and their automorphism groups.

The notion of a projective system was introduced by Tsfasman and Vlăduţ as an algebraic geometric counterpart of linear codes (see [20, p. 67] for relevant historical and bibliographical information). An [n, k]_{q}-projective system is just a collection, say P, of n not necessarily distinct points in ℙ^{k−1} = ℙ^{k−1}(𝔽_{q}). The (semilinear) automorphisms of P are projective semilinear isomorphisms (known in classical literature as collineations) g : ℙ^{k−1} → ℙ^{k−1} that preserve P (together with the multiplicities). These form a group that we denote by Aut(P) and call the automorphism group of P; it is a subgroup of the projective semilinear group PΓL(k, 𝔽_{q}). An [n, k]_{q}-projective system P corresponds to an [n, k]_{q}-linear code C and Aut(P) corresponds to Aut(C); more precisely, Aut(C) is a central extension of Aut(P) by the subgroup 𝔽_{q}^{×} of scalar matrices in GL(k, 𝔽_{q}). Likewise, the projective linear isomorphisms (known in classical literature as projectivities) among Aut(P) form a subgroup MAut(P) of PGL(k, 𝔽_{q}) that corresponds to MAut(C). Many, but perhaps not all, of the notions and results of classical coding theory can be translated in the language of projective systems and we refer to the book [20] for the current state of art. It may be noted, in particular, that a “Lang-like” problem [20, Problem 1.1.9] in this book asks to rewrite existing books on coding theory in terms of projective systems, and it is mentioned that the authors consider this to be a rather important and interesting research problem. In the course of our exposition here, we take a small step toward solving this problem by considering automorphisms in the setting of projective systems, and as a dividend, give a short and simple proof of the classical theorem of MacWilliams mentioned earlier.

Now let us return to a description of our main results. First, we deduce from a theorem of Chow that for the projective system P = G(ℓ, m) corresponding to Grassmann codes, Aut(P) is as follows. If m ≠ 2ℓ, then Aut(P) is precisely PΓL(m, 𝔽_{q}) = PΓL(𝔽_{q}^{m}), viewed as a subgroup of PΓL(k, 𝔽_{q}) = PΓL(∧^{ℓ}𝔽_{q}^{m}) by identifying an m × m nonsingular matrix with its ℓ^{th} compound matrix, or in other words, by identifying a linear map f : 𝔽_{q}^{m} → 𝔽_{q}^{m} with its ℓ^{th} exterior power ∧^{ℓ}f : ∧^{ℓ}𝔽_{q}^{m} → ∧^{ℓ}𝔽_{q}^{m}. On the other hand, if m = 2ℓ, then one has to reckon with a correlation and in this case, Aut(P) is a semidirect product of PΓL(m, 𝔽_{q}) with ℤ/2ℤ. We determine the corresponding group Aut(C(ℓ, m)) via central extensions, or more precisely, using group cohomology and certain subgroups of roots of unity in 𝔽_{q}^{×}; see Theorems 3.7 and 3.9 for a more precise statement. Further, one obtains MAut(P) essentially by replacing PΓL with PGL in the above description. Next, we take up the case of affine Grassmann codes. To this end, we analyze the automorphisms of the so-called big cell of the Grassmannian, and show that they can be lifted to automorphisms of the full Grassmannian and the lifts preserve a certain stratification of the complementary Schubert divisor (Theorem 4.2). Using this and the characterization of the automorphisms of Grassmannians, we can conclude that the semilinear and the monomial automorphism groups of affine Grassmann codes are essentially given by a maximal parabolic subgroup of GL(V) modulo scalars, although one has to again make a distinction between the cases m ≠ 2ℓ and m = 2ℓ. (See Theorems 4.5 and 4.6). Finally, we consider the codes associated to Schubert divisors in Grassmannians; these are a special case of Schubert codes introduced in [7] and are in a sense complementary to affine Grassmann codes. We show, in fact, that the automorphism groups of these codes are isomorphic to that of affine Grassmann codes, and that the automorphisms of Schubert divisors in a Grassmannian can be identified with the automorphisms of the big cell of the Grassmannian. However, proving this in the setting of an arbitrary ground field (and in particular, a finite field) seems quite nontrivial and a number of auxiliary results are needed. In particular, we require a classification of the maximal linear subspaces of Schubert divisors. In fact, we determine the maximal linear subspaces of the Schubert divisor as well as those of its big cell in Lemmas 5.1, and 5.3, and these may be of independent interest.

A key ingredient in our determination of the automorphism group of Grassmann codes is a classical result of Chow [6] (Theorem 3.6) that characterizes linepreserving bijections of Grassmannians and can be viewed as a remarkable generalization of the fundamental theorem of projective geometry. This result has also been rediscovered by others (e.g., Nemitz [15]) and it has been extended to the case when the base field is replaced by a division ring (see, e.g., [22]). For a proof of Chow’s theorem, one may refer to the books of Pankov [17] and Wan [22]. The result mentioned earlier about the automorphisms of the projective system P = G(ℓ, m) corresponding to Grassmann codes is, in fact, just a paraphrasing of Chow’s theorem. However, the corresponding result about the automorphisms of Schubert divisors in Grassmannians appears to be new.

2. Projective Systems and Automorphisms of Codes

In this section we consider the notions of equivalence and automorphisms of linear codes from the point of view of projective systems. The purpose is to settle notations and terminology used throughout this paper. For a more leisurely treatment one may refer to [20]. Fix a prime power q and let F denote the finite field 𝔽_{q}. Let C ⊂ F^{n} denote a nondegenerate [n, k]_{q}-linear code. The standard basis of F^{n} will be denoted {e_{1}, …, e_{n}} and the associated dual basis will be denoted {e^{1}, …, e^{n}}. The nondegeneracy of C implies that the restrictions of the functionals e^{1}, …, e^{n} to C are nonzero and they span C^{∗}. Thus if we let P denote the collection of n points of ℙ(C^{∗}) corresponding to the restrictions e_{|C}^{1}, ⋯, e_{|C}^{n}, then P consists of n (not necessarily distinct) points in ℙ(C^{∗}) not lying in a hyperplane of the projective space ℙ(C^{∗}). We say that P ⊂ ℙ(C^{∗}) is the projective system associated with C. Note that dim(C^{∗}) = k and thus ℙ(C^{∗}) may be identified with ℙ_{𝔽_{q}}^{k−1}.

If D ⊂ C is an r-dimensional subcode of C, then the subspace of C^{∗} consisting of functionals vanishing on D has codimension r. Let L_{D} ⊂ ℙ(C^{∗}) be its projectivization, where, by convention, L_{C} = ∅. Using this correspondence D ↦ L_{D} we identify r-dimensional subcodes of C with codimension r subspaces of ℙ(C^{∗}). The weight of an r-dimensional subcode D, denoted ∥D∥, is the cardinality |P\L_{D}|. In other words, ∥D∥ is the number of elements of P (counting multiplicities) that are not in L_{D}. The weight of a nonzero codeword v ∈ C, denoted ∥v∥, is the weight of the one-dimensional subcode generated by it. We recall the following elementary formula [20, p. 7] relating the weight of a subcode to that of its one-dimensional subcodes:

(1)  ∥D∥ = \frac{1}{q^{r−1}} ∑_{[v]∈ℙ(D)} ∥v∥.

Geometrically, ∥D∥ = |P \L_{D}|, and P \L_{D} = ∪_{H⊃L_{D}}P \H, where the union is over hyperplanes H in ℙ(C^{∗}) containing L_{D}. The formula (1) now follows from the fact that for each x ∈ P \L_{D}, there are q^{r−1} hyperplanes containing L_{D} but not x.

We recall the notion of a semilinear map f : V → V′ between vector spaces over any field 𝕂. We say f is μ-semilinear if there is a field automorphism μ of 𝕂 with f(u + cv) = f(u) + μ(c)f(v) for all u, v ∈ V and c ∈ 𝕂. In particular an invertible μ-semilinear transformation of 𝕂^{m} has the form x ↦ A μ(x) where A ∈ GL(m, 𝕂) and μ acts on each entry of x. The group of semilinear automorphisms of a vector space V will be denoted ΓL(V), and also by ΓL(m, 𝕂) when V = 𝕂^{m}. For a semilinear map f : V → V′, we denote the corresponding projective semilinear map from ℙ(V) to ℙ(V′) by [f]. By a projective semilinear isomorphism from ℙ(V) to ℙ(V′) we mean the map induced by a semilinear isomorphism from V to V′. We will denote the group of projective semilinear isomorphisms from ℙ(V) onto itself by PΓL(V), and we note that PΓL(V) is the factor group of ΓL(V) by the subgroup 𝕂^{×} of transformations of V → V of the form v ↦ cv as c varies over the nonzero elements of 𝕂. The semilinear isomorphisms of finite dimensional projective spaces are characterized by the following (see, e.g., [1, Theorem 2.26]):

Theorem 2.1 (Fundamental Theorem of Projective Geometry). Let n ≥ 3. The group of bijective self maps of the projective space ℙ(𝕂^{n}) which take lines to lines, is the group PΓL(n, 𝕂).

We will also need the notion of the transpose of a μ-semilinear map f : V → V′ between vector spaces over any field 𝕂. The transpose f^{∗} : V^{′∗} → V^{∗} is the unique semilinear map satisfying the property:

⟨α, f(v)⟩ = μ(⟨f^{∗}α, v⟩) for all α ∈ V^{′∗} and v ∈ V,

where ⟨ , ⟩ denotes the natural pairing V^{∗} × V → 𝕂. If f is represented as x ↦ Aμ(x) in coordinates with respect to some choice of bases of V and V′, then the transpose is represented in coordinates with respect to the dual bases by ξ ↦ μ^{−1}(A^{t}ξ) where A^{t} is the usual transpose and μ^{−1} acts entry-wise on matrices. For a projective semilinear isomorphism g : ℙ(V) → ℙ(V′), we define a transpose as follows. Let f : V → V′ be any semilinear map such that [f] = g, and let f^{∗} : V^{′∗} → V^{∗} be its transpose. Then the projective map [f^{∗}] : ℙ(V^{′∗}) → ℙ(V^{∗}) induced by f^{∗} is independent of the choice of f and it will be denoted by g∗ and called the transpose of g.

We recall (from [20]) the notions of equivalence and isomorphisms for codes and for projective systems. We allow all maps to be semilinear.

Definition 2.2. Let C, C′ be nondegenerate q-ary linear codes and let P ⊂ ℙ(C^{∗}) and P′ ⊂ ℙ(C^{′∗}) be the corresponding projective systems. We say that P and P′ are equivalent if there is a projective semilinear isomorphism g : ℙ(C^{∗}) → ℙ(C^{′∗}) which carries P to P′. We say that the codes C and C′ are equivalent, and write C ∼ C′, if the corresponding projective systems P and P′ are equivalent.

Note that since the points of P, P′ need not be distinct, when we say g carries P to P′, we mean that for each p ∈ P, the multiplicity of g(p) in P′ equals the multiplicity of p in P. In particular, if two codes are equivalent, then their lengths and dimensions coincide. Suppose both P and P′ consist of n elements, i.e., the length of C as well as C′ is n. Let Isom(F^{n}) be the subgroup of ΓL(n, F) consisting of transformations preserving the Hamming metric on F^{n}. They are of the form x ↦ Aμ(x) where A is a monomial matrix and μ a power of the Frobenius automorphism. If C ∼ C′ with g : ℙ(C^{∗}) → ℙ(C^{′∗}) as above, then there is a unique g^{♭} ∈ Isom(F^{n})/F^{×} that takes C to C′. To see this, let g̃ : C^{∗} → C^{′∗} be a semilinear isomorphism inducing g, then g(P) = P′ implies that g̃ takes e_{|C}^{j} to a_{j} e_{|C′}^{σ(j)} for some permutation σ ∈ S_{n} and some a_{j} ∈ F^{×}. If g̃ is μ-semilinear, then there is a unique μ-semilinear automorphism ϕ^{∗} of (F^{n})^{∗} whose restriction to C^{∗} is g̃. It is given by ϕ^{∗}(ξ) = Aμ(ξ) where A is the monomial matrix in GL(n, F) whose (i, j)th entry is a_{j}δ_{i,σ(j)}. Thus (g̃^{−1})^{∗} : C → C′ is the restriction to C of the isometry ϕ^{−1} of F^{n} that takes x ↦ Bμ(x) where B = (B_{ij}) ∈ GL(n, F) is given by B_{ij} = (1/a_{j}) δ_{i,σ(j)}. Since g̃ was ambiguous up to a scalar multiple, so is the isometry ϕ^{−1}. The class of ϕ^{−1} in Isom(F^{n})/F^{×} is the desired map g^{♭}.

Definition 2.3. Let C ⊂ F^{n} be a nondegenerate [n, k]_{q}-linear code, and P the corresponding projective system. The automorphism group Aut(C) of the code C is the group of transformations of F^{n} onto itself that take C to C and have the form x ↦ Aμ(x), where A is a monomial matrix in GL(n, F) and μ a field automorphism of F. The automorphism group Aut(P) of the projective system P is the subgroup of PΓL(C^{∗}) of transformations taking P to itself.

We remark that for C and P as above, Aut(C) is a central extension of Aut(P) by F^{×}, i.e., Aut(P) ≃ Aut(C)/F^{×}, where F^{×} is the subgroup of scalar matrices in GL(n, F). The isomorphism between Aut(P) and Aut(C)/F^{×} is given by the correspondence g ↦ g^{♭} described above.

In addition to the equivalence relation ∼ of codes (Definition 2.2), there is another natural notion of equivalence of nondegenerate [n, k]_{q} codes:

Definition 2.4. Let C, C′ be nondegenerate q-ary linear codes. Define C ≈ C′ if there is a weight preserving bijection f : C → C′ carrying r-dimensional subcodes of C to r-dimensional subcodes of C′ for all r ≥ 0.

We now give a geometric proof of the following well-known result of MacWilliams. The proof below seems much shorter than the standard proofs (cf. [14, 5, 23]) and illustrates the advantage of the geometric framework of projective systems.

Theorem 2.5 (MacWilliams). Let C, C′ be nondegenerate q-ary linear codes of dimension ≥ 3. Then: C ∼ C′ if and only if C ≈ C′.

Proof. Suppose C ≈ C′ and let f : C → C′ be as in the Definition 2.4. Clearly, dim C = dim C′ = k (say). Since the correspondence D → L_{D} maps (k − 1)- dimensional subcodes of C to codimension k − 1 subspaces of ℙ(C^{∗}), viz., points of ℙ(C^{∗}), we obtain from f an induced bijective map f_{♯} : ℙ(C^{∗}) → ℙ(C^{′∗}). Given an r-dimensional subcode D of C, the map f carries the set of all (k − 1)-dimensional subcodes of C that contain D into the set of (k −1)-dimensional subcodes of C′ containing f(D). This implies that f_{♯} carries codimension r subspaces to codimension r subspaces. Since k ≥ 3, Theorem 2.1 implies that f_{♯} is a projective semilinear isomorphism. Moreover, from formula (1), it follows that f preserves weights of all subcodes. In particular, since points of ℙ(C^{∗}) correspond to (k − 1)-dimensional subcodes, we see that |{v} ∩ P| = |{f_{♯}(v)} ∩ P′|, for all v ∈ ℙ(C^{∗}). Therefore P ∈ P with multiplicity ν if and only if f_{♯}(P) ∈ P′ with multiplicity ν. Thus the semilinear isomorphism f_{♯} carries P to P′, and so C ∼ C′. For the converse, the discussion preceding Definition 2.3 shows that C ∼ C′ implies the existence of a transformation ϕ^{−1} ∈ Isom(F^{n}) of F^{n} taking C to C′. Since Hamming isometries preserve weights of codewords, ϕ_{|C} gives the equivalence C ≈ C′. □

We observe that if C ≈ C′ with f : C → C′ a linear isometry, then the map f_{♯} is automatically a projective linear isomorphism, and hence we do not require the k ≥ 3 assumption necessary for Theorem 2.1. We state this as a corollary:

Corollary 2.6. If f : C → C′ is a linear isometric isomorphism between two q-ary linear codes, then C and C′ have the same length n and f is the restriction to C of a linear transformation of F^{n} defined by a monomial matrix in GL(n, F).

3. Grassmann Codes

In the first subsection below, we work over an arbitrary field 𝕂. Subsequently, we will let 𝕂 be the finite field F = 𝔽_{q}. Fix positive integers ℓ, m. Let G(ℓ, m) denote the set of ℓ-dimensional subspaces of 𝕂^{m}. We will also use the symbol V for 𝕂^{m}, and denote G(ℓ, m) by G_{ℓ}(V). Let {e_{1}, …, e_{m}} be a basis of 𝕂^{m} and let I(ℓ, m) denote the set of multi-indices:

I(ℓ, m) = {(i_{1}, ⋯, i_{ℓ}) : 1 ≤ i_{1} < i_{2} < ⋯ < i_{ℓ} ≤ m}.

For I = (i_{1}, ⋯, i_{ℓ}) ∈ I(ℓ, m), let e_{I} denote the decomposable element e_{i_{1}} ∧⋯∧e_{i_{ℓ}} of ∧^{ℓ}V. Then {e_{I} : I ∈ I(ℓ, m)} is a basis of ∧^{ℓ}V. To any γ ∈ G(ℓ, m), we assign the wedge product v_{1} ∧ v_{2} ∧ ⋯ ∧ v_{ℓ} ∈ ℙ(∧^{ℓ}V) where {v_{1}, …, v_{ℓ}} is an arbitrary basis of γ. Writing v_{1} ∧ v_{2} ∧ ⋯ ∧ v_{ℓ} = ∑_{I} p_{I}e_{I}, with p_{I} ∈ 𝕂 for I ∈ I(ℓ, m), we obtain the Plücker coordinates (p_{I})_{I∈I(ℓ,m)} of γ. It is well-known that the resulting map G(ℓ, m) → ℙ(∧^{ℓ}V) is a nondegenerate embedding of G(ℓ, m) as a projective subvariety of ℙ(∧^{ℓ}V) defined by certain quadratic polynomials (see, e.g., [12, §VII.2]). We will refer to G(ℓ, m) as the Grassmannian or the Grassmann variety (of ℓ-dimensional subspaces of V = 𝕂^{m}). Note that if ℓ = m, then G(ℓ, m) reduces to a point, whereas if ℓ = 1 (or ℓ = m − 1), then G(ℓ, m) is just the projective space ℙ(V). With this in view, to avoid trivialities, we shall henceforth assume that 1 < ℓ < m. This means, in particular, that m ≥ 3.

3.1. Line-preserving Bijections of Grassmannians. By a line in G_{ℓ}(V) we mean a set of ℓ-dimensional spaces of V containing a given (ℓ − 1)-dimensional subspace and contained in a given (ℓ + 1)-dimensional subspace (see the discussion following Lemma 3.5). We define: Aut(G_{ℓ}(V)) to be the set of all bijections f : G_{ℓ}(V) → G_{ℓ}(V) such that both f and f^{−1} take lines to lines. Evidently Aut(G_{ℓ}(V)) is a group with respect to composition. The group Aut(G_{ℓ}(V)) was explicitly determined by Wei-Liang Chow (1949); see Theorem 3.6 below. We will need a few definitions and lemmas (which are also needed in later sections) before we can state this theorem. Note that the assumption ℓ < m is crucial in the lemma below.

Lemma 3.1. Let ρ̂: ΓL(V) → ΓL(∧^{ℓ}V) be the ℓ-th exterior power representation that maps f ∈ ΓL(V) to ∧^{ℓ}f, where ∧^{ℓ}f ∈ ΓL(∧^{ℓ}V) is defined by

(∧^{ℓ}f)(v_{1} ∧ ⋯ ∧ v_{ℓ}) = f(v_{1}) ∧ ⋯ ∧ f(v_{ℓ}) for v_{1}, …, v_{ℓ} ∈ V.

Then the induced homomorphism ρ : PΓL(V) → PΓL(∧^{ℓ}V) is injective. Moreover, the kernel of ρ̂ is

ker(ρ̂) = {c I_{m} : c ∈ 𝕂 with c^{ℓ} = 1}.

Proof. For γ ∈ G_{ℓ}(V) and g ∈ PΓL(V), denote by g(γ) ∈ G_{ℓ}(V) the image under g of the subspace γ of V. Let g ∈ ker(ρ). Then for any γ ∈ G_{ℓ}(V), the Plücker coordinates of g(γ) and γ are equal, and so g(γ) = γ. Suppose there exists v ∈ ℙ(V) with g(v) ≠ v. Since ℓ < m, we can find γ ∈ G_{ℓ}(V) containing v such that g(v) ∉ γ. But v ∈ γ implies g(v) ∈ g(γ) = γ, which is a contradiction. This shows that g is the identity element of PΓL(V). Thus ρ is injective. It follows that the elements of ker(ρ̂) are scalar matrices, and clearly c I_{m} ∈ ker(ρ̂) if and only if c^{ℓ} = 1. □

We recall that the group PGL(m, 𝕂) has an inverse transpose automorphism taking [A] ↦ [A^{−t}], where A ∈ GL(m, 𝕂) is a representative of [A]. For m ≥ 3, this is an outer automorphism corresponding to a diagram automorphism of the root system. (For m = 2, the automorphism is inner because [A^{−t}] = [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] [A] [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ]^{−1}). There is an induced automorphism on ΓL(V) taking a μ-semilinear map x ↦ Aμ(x) to the μ-semilinear map x ↦ A^{−t}μ(x), which in turn descends to an automorphism of PΓL(V). We denote these automorphisms of ΓL(V) and PΓL(V) by a common symbol −t, and refer to it as the inverse transpose outer automorphism.

Definition 3.2. The Hodge star isomorphism ∗_{ℓ} : ∧^{ℓ}V → ∧^{m−ℓ}V is the linear isomorphism of ∧^{ℓ}V defined by e_{I} → sgn(II^{∘}) e_{I^{∘}} where I ∈ I(ℓ, m) and I^{∘} ∈ I(m − ℓ, m) is the complement of I. The scalar sgn(II^{∘}) ∈ {±1} denotes the sign of the permutation (1, ⋯, m) ↦ (II^{∘}). We use the same symbol ∗_{ℓ} : ℙ(∧^{ℓ}V) → ℙ(∧^{m−ℓ}V) for the induced projective linear isomorphism.

Proposition 3.3. Let V = 𝕂^{m}. The isomorphism ∗_{ℓ} takes G_{ℓ}(V) to G_{m−ℓ}(V). Moreover, ∗_{ℓ} ∘ ∧^{ℓ}[f] ∘ ∗_{ℓ}^{−1} = ∧^{m−ℓ}[f]^{−t} for all [f] ∈ PΓL(V). In the case m = 2ℓ, ∧^{ℓ}[f] ↦ ∗_{ℓ} ∘ ∧^{ℓ}[f] ∘ ∗_{ℓ}^{−1} is a nontrivial outer automorphism of im(ρ).

Proof. Let f ∈ ΓL(V) be a representative of [f], and let x ↦ Aμ(x) denote the coordinate expression of f with respect to the basis {e_{1}, …, e_{m}}. We will show ∗_{ℓ} ∘∧^{ℓ}f ∘∗_{ℓ}^{−1} = det(A)∧^{m−ℓ} f^{−t} holds ΓL(V), which in turn implies that ∗_{ℓ} ∘∧^{ℓ}[f]∘ ∗_{ℓ}^{−1} = ∧^{m−ℓ}[f]^{−t}. We construct linear isomorphisms H_{ℓ} : ∧^{ℓ}V → ∧^{m−ℓ}V^{∗} and θ : V^{∗} → V such that ∗_{ℓ} = (∧^{m−ℓ}θ) ∘ H_{ℓ} : ∧^{ℓ}V → ∧^{m−ℓ}V. Let η = e_{1} ∧ ⋯ ∧ e_{m} ∈ ∧^{m}V. For any ξ ∈ ∧^{ℓ}V, let H_{ℓ}(ξ) ∈ ∧^{m−ℓ}V^{∗} be defined by ⟨H_{ℓ}(ξ), ζ⟩ η = ξ ∧ ζ for all ζ ∈ ∧^{m−ℓ}V, where ⟨, ⟩ is the natural pairing between ∧^{i}V^{∗} and ∧^{i}V for each 0 ≤ i ≤ m. We claim that H_{ℓ} carries a decomposable element v_{1} ∧ ⋯ ∧ v_{ℓ} of ∧^{ℓ}V to a decomposable element of ∧^{m−ℓ}V^{∗}. To see this, we extend {v_{1}, …, v_{ℓ}} to a basis {v_{1}, …, v_{m}} of V such that v_{1} ∧ ⋯ ∧ v_{m} = η. It follows from the definition of H_{ℓ} that H_{ℓ}(v_{1} ∧ ⋯ ∧ v_{ℓ}) is the decomposable element v^{ℓ+1} ∧ ⋯ ∧ v^{m}, where {v^{1}, …, v^{m}} is the dual basis. In particular H_{ℓ}(e_{I}) = sgn(II^{∘})e^{I^{∘}}. Now let θ : V^{∗} → V be the linear isomorphism defined by e^{i} ↦ e_{i} for all i. It now follows that (∧^{m−ℓ}θ) ∘ H_{ℓ} carries e_{I} to sgn(II^{∘}) e_{I^{∘}}, and hence (∧^{ℓ}θ) ∘ H_{ℓ} is ∗_{ℓ}. The map ∧^{m−ℓ}θ takes a decomposable element v^{1} ∧ ⋯ ∧ v^{m−ℓ} ∈ ∧^{m−ℓ}V^{∗} to the decomposable element θ(v^{1}) ∧ ⋯ ∧ θ(v^{m−ℓ}). Thus the composition (∧^{m−ℓ}θ) ∘ H_{ℓ} takes decomposable elements to decomposable elements . Therefore the projective map ∗_{ℓ} takes G_{ℓ}(V) to G_{m−ℓ}(V).

Next, we observe that the maps f^{−t} and θ ∘ (f^{−1})^{∗} ∘ θ^{−1} are equal, as both have the coordinate expression x ↦ A^{−t}μ(x) (the transpose f^{∗} of a semilinear map f was defined immediately after Theorem 2.1). Using the definition ∗_{ℓ} = (∧^{m−ℓ}θ)∘H_{ℓ} we get, ∗_{ℓ} ∘ ∧^{ℓ}f ∘ ∗_{ℓ}^{−1} = ∧^{m−ℓ}θ ∘ H_{ℓ} ∘ ∧^{ℓ}f ∘ H_{ℓ}^{−1} ∘ ∧^{m−ℓ}θ^{−1}. Therefore, it suffices to show H_{ℓ} ∘ ∧^{ℓ}f ∘ H_{ℓ}^{−1} = det(A) ∧^{m−ℓ} (f^{−1})^{∗}. To prove this we need to show:

(2)  ⟨(H_{ℓ} ∘ ∧^{ℓ}f)(ξ), ζ⟩ = det(A) ⟨(∧^{m−ℓ}(f^{−1})^{∗} ∘ H_{ℓ})(ξ), ζ⟩

holds for all ξ ∈ ∧^{ℓ}V, ζ ∈ ∧^{m−ℓ}V. Multiplying both sides of (2) by η, the left side is (∧^{ℓ}f(ξ)) ∧ ζ which we rewrite as:

(3)  ∧^{m}f (ξ ∧ (∧^{m−ℓ}f^{−1})(ζ)) = μ (⟨H_{ℓ}(ξ), ∧^{m−ℓ}f^{−1}(ζ)⟩) det(A) η

which is η multiplied by the right side of (2), as desired.

Finally, suppose m = 2ℓ. Note that m ≥ 3 since ℓ > 1. Assume, on the contrary, that ∧^{ℓ}[f] ↦ ∗_{ℓ} ∘ ∧^{ℓ}[f] ∘ ∗_{ℓ}^{−1} is an inner automorphism of im(ρ), obtained by conjugating by ∧^{ℓ}[f_{0}]. Then, using ∗_{ℓ}∘∧^{ℓ}[f]∘∗_{ℓ}^{−1} = ∧^{ℓ}[f]^{−t}, we get ρ([f_{0}∘f∘f_{0}^{−1}]) = ρ([f]^{−t}). Since ρ is injective, we get a contradiction to the fact that −t is not an inner automorphism of PΓL(V). □

Corollary 3.4. Assume that m = 2ℓ. Consider the semidirect product

PΓL(V) ⋊_{−t} ℤ/2ℤ := ⟨PΓL(V), ϵ | ϵ^{2} = I_{PΓL(V)}, ϵgϵ^{−1} = g^{−t}⟩.

Let im(ρ) ⋊_{∗_{ℓ}} ℤ/2ℤ be the subgroup of PΓL(∧^{ℓ}V) generated by im(ρ) and ∗_{ℓ}:

im(ρ) ⋊_{∗_{ℓ}} ℤ/2ℤ := ⟨im(ρ), ∗_{ℓ} | ∗_{ℓ}^{2} = I_{PΓL(∧^{ℓ}V)}, ∗_{ℓ}(∧^{ℓ}g)∗_{ℓ}^{−1} = ∧^{ℓ}g^{−t}, ⟩.

where I_{G} denotes the identity element of a group G. The homomorphism from PΓL(V) ⋊_{−t} ℤ/2ℤ to im(ρ) ⋊_{∗_{ℓ}} ℤ/2ℤ that restricts to ρ on PΓL(V) and sends the generator ϵ to ∗_{ℓ} is an isomorphism.

Proof. We note that ∗_{ℓ}^{2} = (−1)^{ℓ(m−ℓ)}I_{ΓL(∧^{ℓ}F^{m})}, and hence when m = 2ℓ we get ∗_{ℓ}^{2} = (−1)^{ℓ}I_{ΓL(∧^{ℓ}V)} in ΓL(∧^{ℓ}V) and ∗_{ℓ}^{2} = I_{PΓL(∧^{ℓ}V)} in PΓL(∧^{ℓ}V). Therefore the homomorphism described in the statement is well defined. It is clearly surjective, and it is injective because ρ is injective and ∗_{ℓ} ∉ im(ρ) (because ∧^{ℓ}g ↦ ∗_{ℓ}∘∧^{ℓ}g∘∗_{ℓ}^{−1} is not an inner automorphism of im(ρ)). □

Remark: We will also need the following relation between ∗_{ℓ−1} and ∗_{ℓ}, which follows from their definitions.

(4)  ∗_{ℓ} (β ∧ v) = ι_{θ^{−1}(v)}(∗_{ℓ−1}β) for all β ∈ ∧^{ℓ−1}V, v ∈ V,

where ι_{ω} : ∧^{r}V → ∧^{r−1}V is the interior multiplication operation defined by ⟨ν, ι_{ω}ξ⟩ = ⟨ω ∧ ν, ξ⟩ for all ν ∈ ∧^{r−1}V^{∗}, ω ∈ V^{∗} and ξ ∈ ∧^{r}V.

We use the letters α, β, γ and δ for points of G_{ℓ−2}(V), G_{ℓ−1}(V), G_{ℓ}(V), and G_{ℓ+1}(V) respectively. By a linear subspace of G_{ℓ}(V), we mean a linear subspace of ℙ(∧^{ℓ}V) contained in G_{ℓ}(V). The following basic fact characterizes the linear subspaces of G_{ℓ}(V) (cf. [6, §II.1], [11, §24.2], [10, p. 68], [8, §2]).

Lemma 3.5 (Maximal linear subspaces of G_{ℓ}(V)). For any β ∈ G_{ℓ−1}(V) and any δ ∈ G_{ℓ+1}(V), let

π_{β} = {γ ∈ G_{ℓ}(V) : γ ⊃ β} and π^{δ} = {γ ∈ G_{ℓ}(V) : γ ⊂ δ}.

Then π_{β} and π^{δ} are linear subspaces of G_{ℓ}(V), of dimensions m − ℓ and ℓ respectively. Moreover, any linear subspace of G_{ℓ}(V) is contained in a π_{β} or a π^{δ}.

We note that the intersection of all γ contained in a π_{β} is β. Therefore, β → π_{β} is one-one. Similarly the vector space sum of all γ in a π^{δ} is δ, therefore δ → π^{δ} is one-one. In case m = 2ℓ, both π_{β} and π^{δ} are ℓ-dimensional, but π_{β} ≠ π^{δ} because the intersection of all γ in a π_{β} is β, where as it is just {0} for a π^{δ}. We note that π^{δ} and π_{β} are projectively isomorphic to ℙ(δ^{∗}) and ℙ(V/β) respectively. Therefore the linear subspaces of π^{δ} and π_{β} (of codimension r) are:

π_{W}^{δ} := {γ ∈ G_{ℓ}(δ) : γ ⊃ W } for W ∈ G_{r}(δ), and (5)

π_{β}^{U} := {γ ∈ G_{ℓ}(V) : β ⊂ γ ⊂ U} for U ∈ G_{m−r}(V) with β ⊂ U,

respectively. In particular, the lines contained in a π^{δ} or in a π_{β} are of the form π_{β}^{δ} for β ∈ G_{ℓ−1}(δ). It follows that every line in G_{ℓ}(V) is of the form π_{β}^{δ}. We also note that π_{β} ∩ π^{δ} is the line π_{β}^{δ} if β ⊂ δ, and is empty otherwise. Similarly π_{β} ∩ π_{β′} is the point β + β′ or empty depending on whether β, β′ are collinear in G_{ℓ−1}(V) or not, and π_{δ} ∩ π_{δ′} is the point δ ∩ δ′ or empty depending on whether δ, δ′ are collinear in G_{ℓ+1}(V) or not. We now state an important theorem of Chow which will be used in the sequel.

Theorem 3.6 (Chow). Let V = 𝕂^{m} with m ≥ 3. Then

Aut(G_{ℓ}(V)) = \begin{cases}im(ρ) ≃ PΓL(V) & if m ≠ 2ℓ, \\ im(ρ) ⋊_{∗_{ℓ}} ℤ/2ℤ ≃ PΓL(V) ⋊_{−t} ℤ/2ℤ & if m = 2ℓ. \end{cases}

Proofs of this theorem are given in Chow [6, §II.1], Pankov [17, Theorem 3.2], Wan [22, Theorem 3.45] and Nemitz [15], with two differences. The first is that Aut(G_{ℓ}(V)) is defined, in these works, to be the group of bijections f : G_{ℓ}(V) → G_{ℓ}(V) such that f, f^{−1} take collinear points to collinear points. It follows from the proof that all such f, f^{−1} also take lines to lines. Another difference is that in the case m = 2ℓ, if f ∈ Aut(G_{ℓ}(V)) is not in im(ρ) then the proofs in these works establish that f is induced by a correlation, i.e., there is a semilinear isomorphism θ̃ : V^{∗} → V such that f is (the restriction to G_{ℓ}(V) of) the projectivization of ∧^{ℓ}θ̃ ∘ H_{ℓ} . We recall the maps H_{ℓ}, θ and ∗_{ℓ} defined in Proposition 3.3 and its proof. Since θ^{−1} ∘ θ̃ ∈ ΓL(V), the full group Aut(G_{ℓ}(V)) (when m = 2ℓ) is the subgroup of PΓL(∧^{ℓ}V) generated by im(ρ) and ∗_{ℓ}, the structure of which is given in Corollary 3.4.

3.2. Automorphism group of C(ℓ, m). Here, and in the remainder of this subsection, we take 𝕂 to be the finite field F = 𝔽_{q} with q elements. In this case G(ℓ, m) is a finite set. We will denote the number \binom{m}{ℓ} as k, and the cardinality |G(ℓ, m)| as n. The projective system P = G(ℓ, m) ⊂ ℙ(∧^{ℓ}F^{m}) gives rise to the [n, k]_{q} Grassmann code C = C(ℓ, m). Let {P_{1}, ⋯, P_{n}} denote representatives in ∧^{ℓ}F^{m} of the n points of G(ℓ, m) taken in some fixed order. The code C is given by

C = {(ω(P_{1}), ⋯, ω(P_{n})) : ω ∈ (∧^{ℓ}F^{m})^{∗}}

The k × n matrix M whose (i, j)-th entry is the i-th Plücker coordinate of the j-th point of P is a generator matrix for C = C(ℓ, m). This requires picking an ordering of the k elements of I_{ℓ,m}. Now Definition 2.3 of the automorphism group gives:

(6)  Aut(P) = {g ∈ PΓL(∧^{ℓ}F^{m}) : g(G(ℓ, m)) = G(ℓ, m)}, and
Aut(C) ≃ π^{−1}(Aut(P)),

where π : ΓL(∧^{ℓ}F^{m}) → PΓL(∧^{ℓ}F^{m}) is the canonical homomorphism.

Theorem 3.7. For the projective system P = G(ℓ, m) ⊂ ℙ(∧^{ℓ}F^{m}), we have:

Aut(P) ≃ \begin{cases}PΓL(m, F) & if m ≠ 2ℓ, \\ PΓL(m, F) ⋊_{−t} ℤ/2ℤ & if m = 2ℓ. \end{cases}

This result is a consequence of Theorem 3.6 (see also Westwick [24]): The group Aut(P) according to (6) consists of those elements of PΓL(∧^{ℓ}F^{m}) which take G(ℓ, m) to itself. Since elements of PΓL(∧^{ℓ}F^{m}) preserve the set of lines in G(ℓ, m), the result follows from Theorem 3.6. □

In the remaining part of this subsection we give a explicit description of the group Aut(C) = π^{−1}(Aut(P)). This is the subgroup of ΓL(∧^{ℓ}F^{m}) generated by im(ρ̂), the scalar matrices F^{×}, and if m = 2ℓ also ∗_{ℓ}. We need some definitions.

Definition 3.8. Let G denote the subgroup of GL(∧^{ℓ}F^{m}) generated by ρ̂(GL(m, F)) and the group F^{×} of scalar matrices. Let λ = (q − 1, ℓ) be the GCD of q − 1 and ℓ, and let λ′ = (q − 1)/λ. Also let μ_{λ} and μ_{λ′} denote, respectively, the group of λ-th and λ′-th roots of unity in F^{×} (identified with the corresponding scalar matrices in GL(m, F)).

If A ∈ μ_{λ}, then clearly ∧^{ℓ}A is the identity matrix. Thus we have a natural epimorphism

ϱ : (GL(m, F)/μ_{λ}) × F^{×} → G given by (A, c) ↦ c (∧^{ℓ}A).

If (A, d) ∈ ker(ϱ) then d (∧^{ℓ}A) is the identity element of GL(∧^{ℓ}F^{m}), which in turn implies that [A] ∈ PGL(m, F) is in ker(ρ). Since ker(ρ) is trivial, we obtain A = cI_{m} for some c ∈ F^{×}/μ_{λ}. Since d(∧^{ℓ}A) is identity, we get d = c^{−ℓ}. Thus

(7)  ker(ϱ) = {(cI_{m}, c^{−ℓ}) : c ∈ F^{×}/μ_{λ}} ⊂ (GL(m, F)/μ_{λ}) × F^{×}.

Therefore, upon letting K denote ker ϱ, we obtain an isomorphism:

(8)  G ≃ ((GL(m, F)/μ_{λ}) × F^{×})/K.

We now describe Aut(C) in terms of G. The group Aut(C) ⊂ ΓL(∧^{ℓ}F^{m}) is generated by G, Aut(F) (acting entrywise on on GL(∧^{ℓ}F^{m})), and also ∗_{ℓ} if m = 2ℓ. Writing g ∈ G as c(∧^{ℓ}A), we see that σ(g) = σ(c) ∧^{ℓ}σ(A) is in G for all σ ∈ Aut(F); hence we have an injection Aut(F) ↪ Aut(G). In the case m = 2ℓ, we see that ∗_{ℓ}∘g∘∗_{ℓ}^{−1} = c det(A)(∧^{ℓ}A^{−t}) is in G. Since ∗_{ℓ}^{2} differs from I_{GL(∧^{ℓ}F^{m})} by a sign factor of (−1)^{ℓ}, it is more convenient to work with ˜∗_{ℓ} = (∧^{ℓ}κ) ∘ ∗_{ℓ}, where κ ∈ GL(m, F) is the matrix with ones on the antidiagonal and zeros elsewhere, i.e it is the matrix of the linear transformation that sends e_{i} ↦ e_{m−i+1}. Using κ = κ^{−1} = κ^{t} and det(κ) = (−1)^{ℓ(2ℓ−1)}, we get ˜∗_{ℓ}^{2} = (∧^{ℓ}κ ∘ ∗_{ℓ})^{2} = ∧^{ℓ}(κκ^{−t}) det(κ)∗_{ℓ}^{2} = I_{GL(∧^{ℓ}F^{m})}. Therefore we have an injection of the group ℤ/2ℤ generated by ˜∗_{ℓ} into Aut(G). We also note that ˜∗_{ℓ} ∉ G, for otherwise ∧^{ℓ}[f] ↦ ∗_{ℓ} ∘ ∧^{ℓ}[f] ∘ ∗_{ℓ}^{−1} would be an inner automorphism of im(ρ), which is not true (Proposition 3.3). The automorphisms of G corresponding to Aut(F) and ˜∗_{ℓ} commute, and hence we have an injection Aut(F) × ℤ/2ℤ ↪ Aut(G). In summary:

(9)  Aut(C) = \begin{cases}G ⋊ Aut(F) & if m ≠ 2ℓ, \\ G ⋊ (Aut(F) × ℤ/2ℤ) & if m = 2ℓ. \end{cases}

In order to clarify the structure of G as a central extension of PGL(m, F) we consider the following exact sequences:

(10)  1 → μ_{λ′}  ⟶^{ı_{1}} GL(m, F)/μ_{λ} ⟶^{ȷ_{1}} PGL(m, F) → 1
(11)  1 → μ_{λ′}  ⟶^{ı_{2}} F^{×} ⟶^{ȷ_{2}} μ_{λ} → 1
(12) 1 → μ_{λ′} × μ_{λ′} ⟶^{ı_{1}×ı_{2}} (GL(m, F)/μ_{λ}) × F^{×} ⟶^{ȷ_{1}×ȷ_{2}} PGL(m, F) × μ_{λ} → 1

where the map ı_{1} is given by d ↦ d^{1/λ}I_{m} (the map d ↦ d^{1/λ} is the inverse of the isomorphism F^{×}/μ_{λ} → μ_{λ′} given by c ↦ c^{λ}), whereas the map ı_{2} is given by d ↦ d^{ℓ/λ} (note that ı_{2} is injective because (ℓ/λ, λ′) = 1). The map ȷ_{1} is the projectivization map induced from GL(m, F) → PGL(m, F). The map ȷ_{2} takes c ↦ c^{λ′}. The exact sequence (12) is obtained from (10)-(11) by taking products in each factor. We now consider the subgroup K ⊂ (GL(m, F)/μ_{λ}) × F^{×}, and the subgroup K′ ⊂ μ_{λ′} × μ_{λ′} defined by:

K′ = {(d, d^{−1}) : d ∈ μ_{λ′}} = {(c^{λ}, c^{−λ}) : c ∈ F^{×}/μ_{λ}} ⊂ μ_{λ′} × μ_{λ′}.

We observe that the map ı_{1} × ı_{2} of (12) carries K′ isomorphically to K. Thus passing to the quotients (μ_{λ′} × μ_{λ′})/K′ ≃ μ_{λ′} and ((GL(m, F)/μ_{λ}) × F^{×})/K ≃ G in (12), we get the following exact sequence:

(13)  1 → μ_{λ′} ⟶^{[ı_{1}×ı_{2}]} G ⟶^{[ȷ_{1}×ȷ_{2}]} PGL(m, F) × μ_{λ} → 1

Thus the exact sequence (13) expresses G as a central extension by μ_{λ′} of PGL(m, F)× μ_{λ}. The process of arriving at (13) from (10)-(11) can be understood in terms of group cohomology. Using the correspondence between central μ_{λ′}-extensions of a group G and elements of H^{2}(G, μ_{λ′}) (with G acting trivially on μ_{λ′}), let α ∈ Z^{2}(PGL(m, F), μ_{λ′}) and β ∈ Z^{2}(μ_{λ}, μ_{λ′}) be cocycles representing (10) and (11). If p_{1} and p_{2} denote the projection homomorphisms from PGL(m, F) × μ_{λ} to PGL(m, F) and μ_{λ} respectively, then the μ_{λ′}-extension of PGL(m, F) × μ_{λ} that corresponds to the cocycle p_{1}^{∗}(α)+p_{2}^{∗}(β) is given by the amalgamated central product of the extensions corresponding to α and β, i.e., ((GL(m, F)/μ_{λ})×F^{×})/K. We now have a complete description of G and (using (9)) of Aut(C) .

Theorem 3.9. Let λ = (q − 1, ℓ), λ′ = (q − 1)/λ, and let μ_{λ} and μ_{λ′} denote, respectively, the group of λ-th and λ′-th roots of unity in F^{×}. Let p_{1}, p_{2} denote the projections of PGL(m, F) × μ_{λ} onto its factors. The group G is a central extension of (PGL(m, F) × μ_{λ}) by μ_{λ′}, corresponding to the class [p_{1}^{∗}(α)] + [p_{2}^{∗}(β)] ∈ H^{2}((PGL(m, F)×μ_{λ}), μ_{λ′}), where [α] ∈ H^{2}(PGL(m, F), μ_{λ′}) and [β] ∈ H^{2}(μ_{λ}, μ_{λ′}) are classes representing the μ_{λ′}-extensions GL(m, F)/μ_{λ} and F^{×} appearing in (10) and (11).

In the special case when (λ, λ′) = 1, the extension F^{×} splits as F^{×} = μ_{λ} × μ_{λ′}. The class [β] = 0, and hence [p_{1}^{∗}(α)]+[p_{2}^{∗}(β)] = [p_{1}^{∗}(α)]. Thus G ≃ (GL(m, F)/μ_{λ})× μ_{λ}. In particular if λ = (q − 1, ℓ) = 1, the group μ_{λ} is trivial, and so it follows that G ≃ GL(m, F). Similarly, if ℓ ≡ 0 (mod q − 1), then μ_{λ} = F^{×} and hence G ≃ PGL(m, F) × F^{×}.

Remark: The monomial automorphism group of the Grassmann C(ℓ, m), viz., MAut(C(ℓ, m)) = Aut(C(ℓ, m))∩GL(∧^{ℓ}F^{m}) is G if m ≠ 2ℓ and G ⋊ℤ/2ℤ if m = 2ℓ, where the semidirect product is as in (9).

4. Affine Grassmann codes

We continue to use the notations of the previous section. In particular, integers ℓ, m are kept fixed throughout and it may be tacitly assumed that 1 < ℓ < m. To begin with, suppose the ground field 𝕂 is the finite field F = 𝔽_{q} with q elements. We recall P = G(ℓ, m) ⊂ ℙ(∧^{ℓ}F^{m}) is the projective system defining the Grassmann code C(ℓ, m). Let H_{0} denote the coordinate hyperplane of ℙ(∧^{ℓ}F^{m}) given by the vanishing of the Plücker coordinate p_{I_{0}} where I_{0} := (m − ℓ + 1, m − ℓ + 2, …, m). The intersection of H_{0} with G(ℓ, m) is a Schubert divisor of G(ℓ, m). It will be denoted as Ω. The complement of Ω, i.e., G(ℓ, m) \H_{0} will be denoted W_{0}. It is the big cell (of dimension ℓ(m − ℓ)) of G(ℓ, m) in its standard cell structure. Let V_{m−ℓ} denote the span of the first m − ℓ standard basic vectors of F^{m}. We can also describe Ω and W_{0} as:

(14)  Ω = {γ ∈ G(ℓ, m) : dim(γ ∩ V_{m−ℓ}) > 0} and
W_{0} = {γ ∈ G(ℓ, m) : dim(γ ∩ V_{m−ℓ}) = 0}.

We consider the projective system P^{𝔸} = W_{0} ⊂ ℙ(∧^{ℓ}F^{m}). It is known that the minimum distance of the Grassmann code C(ℓ, m) is |W_{0}| = q^{ℓ(m−ℓ)} (see [16]). In other words a hyperplane of ℙ(∧^{ℓ}F^{m}) can intersect G(ℓ, m) in at most |G(ℓ, m)| − |W_{0}| = O(q^{ℓ(m−ℓ)−1}) points. As a consequence, we see that the projective system P^{𝔸} ⊂ ℙ(∧^{ℓ}F^{m}) is nondegenerate. The affine Grassmann code C^{𝔸}(ℓ, m) is the code associated with this projective system. The length and dimension of this code are n = q^{ℓ(m−ℓ)} and k = \binom{m}{ℓ} respectively. By construction, it is obtained from the Grassmann code C(ℓ, m) by puncturing on the coordinates corresponding to the points of P \P^{𝔸}. A k × n generator matrix M^{𝔸} for this code can be constructed as follows: We pick some ordering of the k Plücker coordinates {p_{I} : I ∈ I(ℓ, m)}, such that the first of these is p_{I_{0}} (as defined above). We pick some ordering of the n points of W_{0}. The j-th column of the k × n generator matrix M^{𝔸} is the coordinate vector {p_{I}/p_{I_{0}} : I ∈ I(ℓ, m)} (in the chosen order) of the j-th point of W_{0}. The code C^{𝔸}(ℓ, m) is also an example of an algebraic-geometric code C(X, L; P) (see [21, Chapter 3.1]) constructed by evaluating the global sections H^{0}(X, L(D)) of a line bundle L(D) associated with a divisor D of a smooth projective variety X over F. In this construction the global sections are evaluated on some subset P = {P_{1}, …, P_{n}} of F -rational points of X (after choosing an isomorphism of L_{P_{i}} with F) disjoint from the support of D. For the affine Grassmann code, the triple (X, D, P) is (G(ℓ, m), Ω, W_{0}). To see this we note that the vector space generated by {p_{I}/p_{I_{0}} : I ∈ I(ℓ, m)} (viewed as functions on W_{0}) is just the space L(Ω) := {0} ∪ {f ∈ 𝔽_{q}(G(ℓ, m))^{×} : div(f) + Ω ≥ 0}. Therefore the code C(X, L; P) is monomially equivalent to C^{𝔸}(ℓ, m). The affine Grassmann code was introduced in a slightly different but equivalent formulation by Beelen, Ghorpade and Høholdt [2]. For any γ ∈ W_{0}, there is a unique ℓ × (m − ℓ) matrix A such that the rows of the ℓ × m matrix (A | I_{ℓ}) (where I_{ℓ} is the ℓ × ℓ identity matrix) form a basis for γ ([9, p. 193]). The Plücker coordinates {p_{I}/p_{I_{0}} : I ∈ I(ℓ, m)} of a point γ ∈ W_{0} were interpreted in [2], as the set of all r × r minors (for 0 ≤ r ≤ ℓ) of the matrix A described above. In particular, any generator matrix for the code constructed in [2], differs from M^{𝔸} only by a row transformation and a column permutation.

In this section we determine the automorphism group Aut(P^{𝔸}) of the projective system, or equivalently Aut(C^{𝔸}(ℓ, m))/F^{×} (the automorphism group of the code modulo dilations). The group Aut(C^{𝔸}(ℓ, m)) itself is a central F^{×}-extension of Aut(P^{𝔸}). Of more interest in the case of affine Grassmann codes, is the permutation automorphism group, PAut(C^{𝔸}(ℓ, m)). This is the subgroup of the permutation group on the columns of M^{𝔸}, consisting of permutations which preserve the row space of M^{𝔸}. In [3], a subgroup of PAut(C^{𝔸}(ℓ, m)) was identified, and it was remarked that the full group could be larger and it was shown that this subgroup excludes an element (in fact, an involution) of PAut(C^{𝔸}(ℓ, m)) when ℓ = m−ℓ. We show that the subgroup determined in [3] is, in fact, the full group PAut(C^{𝔸}(ℓ, m)) when m ≠ 2ℓ, and that the excluded involution is essentially the only missing ingredient when m = 2ℓ.

4.1. Automorphisms of the big cell of the Grassmannian. In this subsection, the ground field 𝕂 is arbitrary. Let V = 𝕂^{m}, and let V_{m−ℓ} ⊂ V be the subspace spanned by the first m − ℓ standard basic vectors {e_{1}, …, e_{m−ℓ}}. We recall that W_{0} ⊂ G_{ℓ}(V) consists of those γ which are complementary to V_{m−ℓ}. We define:

Aut(W_{0}) = {g ∈ PΓL(∧^{ℓ}V) : g(W_{0}) = W_{0}}.

We introduce a decomposition of G_{ℓ}(V) as W_{0} ∪ W_{1} ∪ ⋯ ∪ W_{ℓ} where:

(15)  W_{i} := {γ ∈ G_{ℓ}(V) : dim(γ ∩ V_{m−ℓ}) = i}, for 0 ≤ i ≤ ℓ.

The following elementary lemma follows from the fact that G_{ℓ}(V) ⊂ ℙ(∧^{ℓ}V) is cut out by quadratic polynomials.

Lemma 4.1. If a line L ⊂ ℙ(∧^{ℓ}V) intersects G_{ℓ}(V) in two distinct points γ, γ′, then either L ⊂ G_{ℓ}(V) or L ∩ G_{ℓ}(V) = {γ, γ′}.

Theorem 4.2. Viewed as subgroups of PΓL(∧^{ℓ}V), we have Aut(W_{0}) ⊂Aut(G_{ℓ}(V)). Moreover, every g ∈ Aut(W_{0}) also satisfies g(W_{i}) = W_{i} for 0 ≤ i ≤ ℓ.

Remark: The theorem is motivated by the special case when 𝕂 is algebraically closed. In this case for any g ∈ PGL(∧^{ℓ}V) with g(W_{0}) = W_{0}, the intersection g(G_{ℓ}(V)) ∩ G_{ℓ}(V) is a closed subvariety of G_{ℓ}(V) containing W_{0}. Since G_{ℓ}(V) is the Zariski closure of W_{0}, it follows that g(G_{ℓ}(V)) ∩ G_{ℓ}(V) = G_{ℓ}(V). For arbitrary fields this reasoning has to be replaced by an argument such as the following.

Proof. The first assertion follows from the second, because G_{ℓ}(V) is the union W_{0} ∪⋯∪W_{ℓ}. We prove the second assertion by induction on i. The base case i = 0 is true by hypothesis. We assume inductively that g(W_{i}) = W_{i} for 0 ≤ i ≤ r − 1, and establish the result for i = r. For any γ ∈ W_{r}, we pick an (ℓ − 1)-dimensional subspace β ⊂ γ such that dim(β ∩ V_{m−ℓ}) = r − 1, and pick u_{1} ∈ V_{m−ℓ} satisfying γ = β ∧ u_{1}. We pick any u_{2} ∉ (β + V_{m−ℓ}) and consider γ′ = β ∧ (u_{1} + u_{2}). The line L joining γ and γ′ has exactly one point in L ∩ W_{r} namely γ, the remaining points of L are in W_{r−1}. Now let h be any one of the two maps g, g^{−1}. Since h ∈ PΓL(V), the image h(L) is a line in ℙ(∧^{ℓ}V), and since h(W_{r−1}) = W_{r−1} we see that h(L)\G_{ℓ}(V) has at most one point. Therefore, by Lemma 4.1, we have h(L) ⊂ G_{ℓ}(V), i.e., h(γ) ∈ G_{ℓ}(V). It remains to prove that h(γ) ∈ W_{r}. To this end we make two observations about h(L): the first is that |h(L)∩W_{r−1}| > 1, because as mentioned above there is atmost one point of h(L) which is not in W_{r−1}. The second observation is that h(L) has atleast one point not in W_{r−1}, because if h(L) ⊂ W_{r−1}, then the fact that h^{−1} takes W_{r−1} to itself would imply that L ⊂ W_{r−1} which is not true. Together these observations imply that h(γ) ∉ W_{r−1} and all the other points of h(L) are in W_{r−1}. Since h(L) is a line on G_{ℓ}(V), we can express it as h(L) = π_{β′}^{δ} for some β′ ∈ G_{ℓ−1}(V) and δ ∈ G_{ℓ+1}(V). Any γ′ ∈ h(L) other than h(γ) is in W_{r−1}, and hence dim(β′ ∩ V_{m−ℓ}) is r − 2 or r − 1 and dim(δ ∩ V_{m−ℓ}) is r − 1 or r. Of these four cases, in the case dim(β′ ∩ V_{m−ℓ}) = r − 1 and dim(δ ∩ V_{m−ℓ}) = r, the line h(L) has exactly one point in W_{r} which must be h(γ) as was to be shown. In the other three cases, either h(L) ⊂ W_{r−1} or |h(L) ∩ W_{r−1}| = 1 which have been observed above to be false. Since γ ∈ W_{r} was arbitrary, we have shown that h(W_{r}) ⊂ W_{r}. Since h was allowed to be either of the maps g, g^{−1}, we find g(W_{r}) = W_{r}. □

We need some more notation before we can describe the subgroup Aut(W_{0}) of Aut(G_{ℓ}(V)). Let P_{m−ℓ,ℓ} be the maximal parabolic subgroup of GL(V) consisting of matrices that carry V_{m−ℓ} to itself. (i.e., matrices whose ℓ × m − ℓ submatrix on the last ℓ rows and first m − ℓ columns is zero). Let P̂_{m−ℓ,ℓ} := P_{m−ℓ,ℓ} ⋊ Aut(𝕂) denote the subgroup of ΓL(V) consisting of transformations that carry V_{m−ℓ} to itself, and let P̂_{m−ℓ,ℓ}/𝕂^{×} be the corresponding subgroup of PΓL(V). In the case when m = 2ℓ, let −t̃ denote the automorphism g ↦ κg^{−t}κ^{−1} = (κgκ^{−1})^{−t} of GL(V) (where κ was defined before (9)). There is an induced automorphism of P GL(V) and PΓL(V), again denoted by −t̃. For g ∈ P_{ℓ,ℓ}, the matrix κgκ^{−1} has a zero for the submatrix on first ℓ rows and last ℓ columns. Therefore (κgκ^{−1})^{−t} is again in P_{ℓ,ℓ}, and hence −t̃ is an automorphism of P_{ℓ,ℓ}/𝕂^{×} and P̂_{ℓ,ℓ}/𝕂^{×}.

Definition 4.3. Let (P_{ℓ,ℓ}/𝕂^{×}) ⋊_{−t̃} ℤ/2ℤ and (P̂_{ℓ,ℓ}/𝕂^{×}) ⋊_{−t̃} ℤ/2ℤ denote the semidirect product of P_{ℓ,ℓ}/𝕂^{×} and P̂_{ℓ,ℓ}/𝕂^{×} respectively, with ℤ/2ℤ where the generator of ℤ/2ℤ acts by the automorphism −t̃.

Continuing further with the case m = 2ℓ, we note that ˜∗_{ℓ} ∈ ΓL(∧^{ℓ}V) permutes (upto sign) the basic vectors {e_{I} : I ∈ I(ℓ, 2ℓ)} of ∧^{ℓ}V, and in particular sends e_{I_{0}} ↦ (−1)^{ℓ(ℓ+1)/2}e_{I_{0}}, where as before I_{0} = (ℓ + 1, …, 2ℓ). Therefore, ˜∗_{ℓ} ∈ PΓL(∧^{ℓ}V) preserves the hyperplane H_{0} ⊂ ℙ(∧^{ℓ}V). Since ˜∗_{ℓ} ∈ Aut(G_{ℓ}(V)), it takes W_{0} = G_{ℓ}(V) \ H_{0} to itself, i.e., ˜∗_{ℓ} ∈ Aut(W_{0}). Now, Aut(G_{ℓ}(V)) is the subgroup of PΓL(∧^{ℓ}V) generated by im(ρ) and ∗_{ℓ}, where we may replace ∗_{ℓ} by ˜∗_{ℓ} because (∗_{ℓ})^{−1}˜∗_{ℓ} = ∧^{ℓ}κ ∈ im(ρ). Writing an element of Aut(G_{ℓ}(V)) as ρ(f) ∘ ˜∗_{ℓ}^{i} with i ∈ {0, 1}, it follows that Aut(W_{0}) is generated by Aut(W_{0}) ∩ im(ρ) and ˜∗_{ℓ}. Since, ∗_{ℓ} ∘∧^{ℓ}f ∘∗_{ℓ}^{−1} = ∧^{ℓ}f^{−t} in PΓL(∧^{ℓ}V) (Proposition 3.3), we get ˜∗_{ℓ} ∘∧^{ℓ}f ∘˜∗_{ℓ}^{−1} = ∧^{ℓ}(κf^{−t}κ^{−1}). We note that ˜∗_{ℓ} ∉ im(ρ), hence ˜∗_{ℓ} ∉ ρ(P̂_{ℓ,ℓ}/𝕂^{×}). We denote the subgroup of PΓL(∧^{ℓ}V) generated by ˜∗_{ℓ} and ρ(P̂_{ℓ,ℓ}/𝕂^{×}) by ρ(P̂_{ℓ,ℓ}/𝕂^{×}) ⋊_{˜∗_{ℓ}} ℤ/2ℤ. Writing:

(P̂_{ℓ,ℓ}/𝕂^{×})⋊_{−t̃} ℤ/2ℤ = ⟨P̂_{ℓ,ℓ}/𝕂^{×}, ϵ | ϵ^{2} = I_{PΓL(m,𝕂)}, ϵfϵ^{−1} = κf^{−t}κ^{−1}⟩, and
ρ(P̂_{ℓ,ℓ}/𝕂^{×})⋊_{˜∗_{ℓ}} ℤ/2ℤ = ⟨ρ(P̂_{ℓ,ℓ}/𝕂^{×}), ˜∗_{ℓ} | ˜∗_{ℓ}^{2} = I_{PΓL(∧^{ℓ}𝕂^{m})}, ˜∗_{ℓ} ∧^{ℓ}f˜∗_{ℓ}^{−1} = ∧^{ℓ}(κf^{−t}κ^{−1})⟩,

it follows that the map from (P̂_{ℓ,ℓ}/𝕂^{×})⋊_{−t̃}ℤ/2ℤ to ρ(P̂_{ℓ,ℓ}/𝕂^{×})⋊_{˜∗_{ℓ}} ℤ/2ℤ that sends f ↦ ∧^{ℓ}f and ϵ ↦ ˜∗_{ℓ} establishes an isomorphism between these groups.

Corollary 4.4. For the big cell W_{0} of the Grassmannian G_{ℓ}(V), we have

Aut(W_{0}) = \begin{cases}ρ(P̂_{m−ℓ,ℓ}/𝕂^{×}) ≃ P̂_{m−ℓ,ℓ}/𝕂^{×} & if m ≠ 2ℓ, \\ ρ(P̂_{ℓ,ℓ}/𝕂^{×}) ⋊_{˜∗_{ℓ}} ℤ/2ℤ ≃ (P̂_{ℓ,ℓ}/𝕂^{×}) ⋊_{−t̃} ℤ/2ℤ & if m = 2ℓ. \end{cases}

Proof. Since any g ∈ P̂_{m−ℓ,ℓ}/𝕂^{×}, by definition carries V_{m−ℓ} to itself, we see that ρ(g) carries W_{0} = {γ ∈ G_{ℓ}(V) : dim(γ ∩ V_{m−ℓ}) = 0} to itself. Conversely, suppose f ∈ im(ρ) is in Aut(W_{0}). Writing f = ∧^{ℓ}g for some g ∈ PΓL(V), we will prove that g(V_{m−ℓ}) = V_{m−ℓ}, and hence conclude that f ∈ P̂_{m−ℓ,ℓ}/𝕂^{×}. To prove this suppose there is a v_{0} ∈ V_{m−ℓ} with g(v_{0}) ∉ V_{m−ℓ}. We pick a γ ∈ G_{ℓ}(V_{m−ℓ}) = W_{ℓ}, such that v_{0} ∈ γ. Since f(γ), i.e., the image of the vector space γ under g contains g(v_{0}) ∉ V_{m−ℓ}, it follows that f(γ) ∉ W_{ℓ}, contradicting the observation in Theorem 4.2, that f carries each W_{i} to itself bijectively, in particular W_{ℓ}. In the case m ≠ 2ℓ, since Aut(G_{ℓ}(V)) is im(ρ) and Aut(W_{0}) ⊂ Aut(G_{ℓ}(V)), we conclude that Aut(W_{0}) = ρ(P̂_{m−ℓ,ℓ}/𝕂^{×}). On the other hand, when m = 2ℓ, we have already observed that Aut(W_{0}) is generated by ˜∗_{ℓ} and Aut(W_{0}) ∩ im(ρ) = ρ(P̂_{ℓ,ℓ}/𝕂^{×}). Hence Aut(W_{0}) = ρ(P̂_{ℓ,ℓ}/𝕂^{×}) ⋊_{˜∗_{ℓ}} ℤ/2ℤ. □

4.2. Permutation Automorphisms of C^{𝔸}(ℓ, m). To begin with, let us consider the semilinear as well as the monomial automorphisms of the affine Grassmann code C^{𝔸}(ℓ, m). By definition, the automorphism group of the projective system P^{𝔸} = W_{0} ⊂ ℙ(∧^{ℓ}F^{m}) is the group Aut(W_{0}) ⊂ PΓL(∧^{ℓ}F^{m}) determined in Corollary 4.4. The group Aut(C^{𝔸}(ℓ, m)) is the subgroup π^{−1}(Aut(W_{0})), where π : ΓL(∧^{ℓ}F^{m}) → PΓL(∧^{ℓ}F^{m}) is the quotient homomorphism. In other words Aut(C^{𝔸}(ℓ, m)) is the subgroup of ΓL(∧^{ℓ}F^{m}) generated by ρ̂(P̂_{m−ℓ,ℓ}) and F^{×} and also ˜∗_{ℓ} if m = 2ℓ. Let G^{A} denote the subgroup of GL(∧^{ℓ}F^{m}) generated by ρ̂(P_{m−ℓ,ℓ}) and F^{×}. We recall from Section 3.2 the epimorphism ϱ : (GL(m, F)/μ_{λ}) × F^{×} → G. Let ϱ_{1} denote the restriction of ϱ to (P_{m−ℓ,ℓ}/μ_{λ}) × F^{×}. The image of ϱ_{1} is G^{A} and since K = ker(ϱ) (see (7)) is contained in (P_{m−ℓ,ℓ}/μ_{λ}) × F^{×}, we have ker(ϱ_{1}) = K as well. Therefore, ϱ_{1} induces an isomorphism

(16)  G^{A} ≃ ((P_{m−ℓ,ℓ}/μ_{λ}) × F^{×})/K.

The group MAut(C^{𝔸}(ℓ, m)) = Aut(C^{𝔸}(ℓ, m)) ∩ GL(∧^{ℓ}F^{m}) is G^{A} if m ≠ 2ℓ and generated by G^{A} and ˜∗_{ℓ} if m = 2ℓ. The group Aut(C^{𝔸}(ℓ, m)) is generated by MAut(C^{𝔸}(ℓ, m)) and Aut(F). Since the automorphisms of Aut(G) induced by Aut(F) and ˜∗_{ℓ} (see the discussion preceding (9)) clearly take G^{A} to itself we have injections Aut(F) ↪ Aut(G), and in case m = 2ℓ, Aut(F) × ℤ/2ℤ ↪ Aut(G). In summary:

Theorem 4.5. The automorphism group of the affine Grassmann code C^{𝔸}(ℓ, m) is given by

Aut(C^{𝔸}(ℓ, m)) = \begin{cases}G^{A} ⋊ Aut(F) & if m ≠ 2ℓ, \\ G^{A} ⋊ (Aut(F) × ℤ/2ℤ) & if m = 2ℓ. \end{cases}

whereas the monomial automorphism group of C^{𝔸}(ℓ, m) is given by

MAut(C^{𝔸}(ℓ, m)) = \begin{cases}G^{A} & if m ≠ 2ℓ, \\ G^{A} ⋊ ℤ/2ℤ & if m = 2ℓ. \end{cases}

Since we are mainly interested in the permutation automorphism group of C^{𝔸}(ℓ, m), we do not further discuss the structure of Aut(C^{𝔸}(ℓ, m)) as a central F^{×}-extension of Aut(W_{0}). We now describe PAut(C^{𝔸}(ℓ, m)).

Theorem 4.6. The permutation automorphism group of the affine Grassmann code C^{𝔸}(ℓ, m) is given (up to isomorphism) by

PAut(C^{𝔸}(ℓ, m)) ≃ \begin{cases}P_{m−ℓ,ℓ}/F^{×} & if m ≠ 2ℓ, \\ P_{ℓ,ℓ}/F^{×} ⋊_{−t̃} ℤ/2ℤ & if m = 2ℓ, \end{cases}

where P_{m−ℓ,ℓ} is the maximal parabolic subgroup of GL(V) of transformations preserving V_{m−ℓ} (the span of the first m − ℓ standard basic vectors of F^{m}), and where F^{×} ⊂ P_{m−ℓ,ℓ} denotes the subgroup of scalar matrices, and −t̃ is the automorphism of P_{ℓ,ℓ} defined in the discussion preceding Definition 4.3.

Proof. PAut(C^{𝔸}(ℓ, m)) is the subgroup of MAut(C^{𝔸}(ℓ, m)) consisting of those automorphisms which preserve the set of vectors in ∧^{ℓ}F^{m} having Plücker coordinate p_{I_{0}} = 1, where I_{0} := (m − ℓ + 1, m − ℓ + 2, …, m). We write a general element of P_{m−ℓ,ℓ} as [ \begin{matrix}A & u \\ 0 & B \end{matrix}], where A ∈ GL(m − ℓ, F), B ∈ GL(ℓ, F) and u is an m − ℓ × ℓ matrix. We first consider the case m ≠ 2ℓ. In this case MAut(C^{𝔸}(ℓ, m)) = G^{A}, and the subgroup PAut(C^{𝔸}(ℓ, m)) is the the subgroup of G_{1}^{A} ⊂ G^{A} consisting of elements which preserve the set of ξ ∈ ∧^{ℓ}F^{m} with p_{I_{0}}(ξ) = 1. Let H := ϱ_{1}^{−1}(G_{1}^{A}). In other words:

(17)  H := \{([ \begin{matrix}A & u \\ 0 & B \end{matrix}] mod μ_{λ}, d) : det(B) = d^{−1}\} ⊂ P_{m−ℓ,ℓ}/μ_{λ} × F^{×}.

By definition of H we see that ϱ_{1} induces an isomorphism of H/K with G_{1}^{A} = PAut(C^{𝔸}(ℓ, m)) (where as before K = ker(ϱ) =ker(ϱ_{1})). Now, the epimorphism from H to P_{m−ℓ,ℓ}/F^{×} given by

([ \begin{matrix}A & u \\ 0 & B \end{matrix}] mod μ_{λ}, det(B)^{−1}) ⟼ [ \begin{matrix}A & u \\ 0 & B \end{matrix}] mod F^{×}

has exactly K for its kernel, thus proving that PAut(C^{𝔸}(ℓ, m)) ≃ P_{m−ℓ,ℓ}/F^{×}.

Next, we consider the case m = 2ℓ. Here, MAut(C^{𝔸}(ℓ, m)) is the subgroup of GL(∧^{ℓ}F^{m}) generated by G^{A} and ˜∗_{ℓ}:

MAut(C^{𝔸}(ℓ, m)) = ⟨ G^{A}, ˜∗_{ℓ} | ˜∗_{ℓ}^{2} = I_{GL(∧^{ℓ}F^{m})}, ˜∗_{ℓ}(c ∧^{ℓ}f)˜∗_{ℓ}^{−1} = c det(f) ∧^{ℓ}(κf^{−t}κ^{−1}) ⟩ .

We consider the group:

G̃ := ⟨P_{ℓ,ℓ}/μ_{λ} × F^{×}, ϵ | ϵ^{2} = I_{P_{ℓ,ℓ}/μ_{λ}×F^{×}}, ϵ(f, c)ϵ^{−1} = (κf^{−t}κ^{−1}, det(f) c)⟩ .

(where κf^{−t}κ^{−1}mod μ_{λ} and det(f) only depend on f mod μ_{λ}, because −t̃ preserves μ_{λ} and λ divides m = 2ℓ). The map ϱ′_{1} : G̃ → MAut(C^{𝔸}(ℓ, m)) that equals ϱ_{1} on P_{ℓ,ℓ}/μ_{λ} × F^{×} and that sends ϵ ↦ ˜∗_{ℓ} is a well defined epimorphism, with ker(ϱ′_{1}) = K again. Since ˜∗_{ℓ} sends e_{I_{0}} ↦ (−1)^{ℓ(ℓ+1)/2}e_{I_{0}}, the subgroup PAut(C^{𝔸}(ℓ, m)) ⊂ MAut(C^{𝔸}(ℓ, m)) is generated by G_{1}^{A} and (−1)^{ℓ(ℓ+1)/2}˜∗_{ℓ}, and hence the group H̃ := (ϱ′_{1})^{−1}(PAut(C^{𝔸}(ℓ, m))) is generated by the group H (defined above) and the element ϵ_{1} := (I_{P_{ℓ,ℓ}/μ_{λ}}, (−1)^{ℓ(ℓ+1)/2})ϵ. By definition of H̃, ϱ′_{1} induces an isomorphism H̃/K ≃ PAut(C^{𝔸}(ℓ, m)). The group H̃ ⊂ G̃ has the presentation:

⟨H, ϵ_{1} | ϵ_{1}^{2} = I_{H}, ϵ_{1} ([ \begin{matrix}A & u \\ 0 & B \end{matrix}] mod μ_{λ}, det(B)^{−1}) ϵ_{1}^{−1} = (−t̃([ \begin{matrix}A & u \\ 0 & B \end{matrix}]) mod μ_{λ}, det(A))⟩ .

Now −t̃([ \begin{matrix}A & u \\ 0 & B \end{matrix}]) = κ [ \begin{matrix}A & u \\ 0 & B \end{matrix}]^{−t} κ^{−1} is of the form [ \begin{matrix}B̃ & ∗ \\ 0 & Ã \end{matrix} ], where Ã_{ij} = (A^{−t})_{ℓ−i+1,ℓ−j+1}.

Since Ã is obtained from A^{−t} by permuting the rows as well as the columns by the same permutation, we see det(Ã) = det(A)^{−1} and hence ϵ_{1}Hϵ_{1}^{−1} = H. The group P_{ℓ,ℓ}/F^{×} ⋊_{−t̃} ℤ/2ℤ of Definition 4.3 has the presentation:

⟨P_{ℓ,ℓ}/F^{×}, ϵ | ϵ^{2} = I_{PGL(m,F)}, ϵfϵ^{−1} = κf^{−t}κ^{−1}⟩ .

The map from H̃ → P_{ℓ,ℓ}/F^{×} ⋊_{−t̃} ℤ/2ℤ that restricts to the epimorphism H → P_{m−ℓ,ℓ}/F^{×} defined above in the case m ≠ 2ℓ, and which sends ϵ_{1} ↦ ϵ is a well defined epimorphism with kernel being K again. Therefore H̃/K and hence PAut(C^{𝔸}(ℓ, m)) are isomorphic to P_{ℓ,ℓ}/F^{×} ⋊_{−t̃} ℤ/2ℤ. □

5. Code associated with the Schubert divisor of G(ℓ, m)

We use the notations of the previous section. Also ℓ, m ∈ ℤ are kept fixed and it will be assumed that 1 < ℓ < m. We recall that the Schubert divisor Ω is the intersection of G(ℓ, m) with the hyperplane H_{0} of ℙ(∧^{ℓ}F^{m}) as defined in Section 4. We throughout assume m ≥ 3, for otherwise Ω is ℙ^{0} or ∅. We consider the projective system P_{Ω} = Ω ⊂ H_{0}. The fact that the second higher weight d_{2} of the Grassmann code C(ℓ, m) is q^{ℓ(m−ℓ)−1}(1 + q) (see [8, 16]) together with the fact that |G_{ℓ}(V)| = q^{ℓ(m−ℓ)} + |Ω|, implies that any codimension 2 subspace of ℙ(∧^{ℓ}F^{m}) intersects G_{ℓ}(V) in at most |Ω| − q^{ℓ(m−ℓ)−1} points. In particular the projective system P_{Ω} ⊂ H_{0} is nondegenerate. The Schubert code C_{Ω}(ℓ, m) is the linear code associated with this projective system. The study of these codes, and of more general Schubert codes, goes back to [7]. It may be noted that in the notation of [7], C_{Ω}(ℓ, m) = C_{α}(ℓ, m), where α ∈ I(ℓ, m) is given by α_{1} = m − ℓ and α_{j} = m − ℓ + j for 2 ≤ j ≤ ℓ. In this section we determine the automorphism group of C_{Ω}(ℓ, m).

5.1. Automorphisms of the Schubert divisor of the Grassmannian. In this subsection the Grassmann variety G(ℓ, m) will be defined over an arbitrary field 𝕂. We define:

Aut(Ω) = {g ∈ PΓL(H_{0}) : g(Ω) = Ω}.

We need a few results before we can determine Aut(Ω). We begin with an analogue of Lemma 3.5 for the Schubert divisor Ω. We continue to use the notation introduced in Lemma 3.5 and in the preceding and succeeding paragraph. In particular, β and γ denote elements of G_{ℓ−1}(V) and G_{ℓ+1}(V), respectively.

Lemma 5.1. The maximal linear subspaces of Ω are

π_{β}  of dimension m − ℓ, where dim(β ∩ V_{m−ℓ}) > 0,
π^{δ}  of dimension ℓ, where dim(δ ∩ V_{m−ℓ}) > 1,
π̃_{β} := π_{β}^{β+V_{m−ℓ}} of dimension m − ℓ − 1, where dim(β ∩ V_{m−ℓ}) = 0,
π̃^{δ} = π_{δ∩V_{m−ℓ}}^{δ} of dimension ℓ − 1, where dim(δ ∩ V_{m−ℓ}) = 1.

Here V_{m−ℓ} is as defined in the beginning of Section 4.

Proof. Let π be a maximal linear subspace of Ω. Since π is also a linear subspace of G_{ℓ}(V), π is contained in some π_{β} ∩ Ω or some π^{δ} ∩ Ω. Suppose π ⊂ π_{β} ∩ Ω. If dim(β ∩V_{m−ℓ}) ≥ 1, then π_{β} ∩Ω = π_{β} and so π = π_{β}, whereas if dim(β ∩V_{m−ℓ}) = 0, then π_{β} ∩ Ω = π_{β}^{β+V_{m−ℓ}} and so π = π̃_{β}. Similarly, suppose π ⊂ π^{δ} ∩ Ω (where dim(δ ∩ V_{m−ℓ}) ≥ 1 necessarily). If dim(δ ∩ V_{m−ℓ}) ≥ 2, then π^{δ} ∩ Ω = π^{δ} and hence π = π^{δ}, whereas if dim(δ ∩V_{m−ℓ}) = 1 then π^{δ} ∩Ω = π_{δ∩V_{m−ℓ}}^{δ} and hence π = π̃^{δ}. □

As before (∩_{γ∈π̃_{β}}γ) = β and (∪_{γ∈π̃^{δ}}) = δ show that β ≠ β′ implies π̃_{β} ≠ π̃_{β′}, and δ ≠ δ′ implies π̃^{δ} ≠ π̃^{δ′}. Moreover in case m = 2ℓ, no π̃_{β} is a π̃^{δ}, although they are both (ℓ − 1)-dimensional. We recall that π_{β} ∩ π_{β′} is the point β + β′ when dim(β + β′) = ℓ, and is empty otherwise. Therefore, π̃_{β} ∩ π̃_{β′} is the point β + β′ when β + β′ ∈ W_{1} ⊂ G_{ℓ}(V), and is empty otherwise. Similarly, π_{δ} ∩ π_{δ′} is the point δ∩δ′ if dim(δ∩δ′) = ℓ, and is empty otherwise. Therefore, π̃_{δ} ∩π̃_{δ′} is the point δ∩δ′ if δ ∩ δ′ ∈ W_{1} ∈ G_{ℓ}(V), and is empty otherwise. Finally, π_{β} ∩ π^{δ} is the line π_{β}^{δ} if β ⊂ δ, and is empty otherwise. Therefore, π̃_{β} ∩ π̃^{δ} is the point β ⊕ (δ ∩ V_{m−ℓ}) ∈ W_{1} when β ⊂ δ, and is empty otherwise. We record this as a lemma for later reference:

Lemma 5.2. Let δ, δ′ ∈ G_{ℓ+1}(V) and β, β′ ∈ G_{ℓ−1}(V). Then:

(i) π̃_{δ} ∩ π̃_{δ′} equals δ ∩ δ′ ∈ W_{1} if dim(δ ∩ δ′) = ℓ, and is empty otherwise. (ii) π̃_{β} ∩ π̃^{δ} equals β ⊕ (δ ∩ V_{m−ℓ}) ∈ W_{1} when β ⊂ δ, and is empty otherwise. (iii) π̃_{β} ∩ π̃_{β′} equals β + β′ when β + β′ ∈ W_{1} ⊂ G_{ℓ}(V), and is empty otherwise.

We recall the decomposition Ω = W_{1} ∪ ⋯ ∪ W_{ℓ} defined in (15). The subset W_{1} is just the Schubert cell associated with the Schubert variety Ω ([9]). The next few results lead to the fact (see Proposition 5.5) that automorphisms of Ω must preserve W_{1}. We define:

W_{0}^{−} = {β ∈ G_{ℓ−1}(V) : dim(β ∩ V_{m−ℓ}) = 0}, and
W_{1}^{+} = {δ ∈ G_{ℓ+1}(V) : dim(δ ∩ V_{m−ℓ}) = 1}.

Lemma 5.3. The maximal linear subspaces of W_{1} are π̃_{β} for β ∈ W_{0}^{−} and π̃^{δ} for δ ∈ W_{1}^{+}. The π̃_{β} are (m − ℓ − 1)-dimensional, and the π̃^{δ} are (ℓ − 1)-dimensional.

Proof. The stated subspaces are maximal by the previous lemma. Conversely, let π be a maximal linear subspace of W_{1}, say of dimension r. It suffices to prove that π is contained in a π̃_{β} or a π̃^{δ}. We can assume r = dim(π) > 0, for otherwise π is a point of W_{1} and every point is contained in some π̃_{β} by the next lemma. By the previous lemma π is contained in a π_{β} ∩ W_{1} or a π^{δ} ∩ W_{1}.

Suppose π ⊂ π_{β} ∩ W_{1}, and let ν_{β} := dim(β ∩ V_{m−ℓ}). If ν_{β} = 0, then π ⊂ π̃_{β}, where as if ν_{β} > 1 then π_{β} ∩ W_{1} = ∅. In the remaining case ν_{β} = 1, writing π = π_{β}^{U} for some U ∈ G_{ℓ+r}(V) containing β, we must have dim(U ∩V_{m−ℓ}) = 1 (for, otherwise π ∩ W_{2} ≠ ∅), and dim(U) ≤ ℓ + 1 (for, otherwise dim(U ∩ V_{m−ℓ}) > 1). Hence dim(U) = ℓ + 1, and we get π ⊂ π̃^{δ} for δ = U.

Similarly, in the case π ⊂ π^{δ}, we consider ν_{δ} := dim(δ ∩ V_{m−ℓ}). Since δ is (ℓ + 1)-dimensional, we must have ν_{δ} ≥ 1. If ν_{δ} = 1, then π ⊂ π̃^{δ}, where as if ν_{δ} > 2 then π^{δ} ∩ W_{1} = ∅. In the remaining case ν_{δ} = 2, we write π = π_{U}^{δ} for some U ∈ G_{ℓ−r}(δ). Now π ∩ W_{2} = ∅ forces dim(U) ≥ ℓ − 1 and dim(U ∩ V_{m−ℓ}) = 0. Since dim(U) = ℓ − r with r > 0, we must have dim(U) = ℓ − 1, and we get π ⊂ π̃_{β} for β = U. □

Lemma 5.4. The Schubert cell W_{1} is the union of all the π̃_{β}’s, as well as the union of all the π̃^{δ}’s. In other words,

⋃_{β∈W_{0}^{−}} π̃_{β} = W_{1} = ⋃_{δ∈W_{1}^{+}} π̃^{δ}.

Proof. By definition, the π̃_{β}’s and the π̃^{δ}’s are contained in W_{1}. Conversely, given any γ ∈ W_{1}, we have dim(γ ∩ V_{m−ℓ}) = 1. Therefore, we can find β ⊂ γ and δ ⊃ γ with β ∈ W_{0}^{−} and δ ∈ W_{1}^{+}. It then follows that γ ∈ π̃_{β} and γ ∈ π̃^{δ}. □

Proposition 5.5. If f ∈ Aut(Ω) then f(W_{1}) = W_{1}.

Remark: The proposition is motivated by the special case when 𝕂 is algebraically closed. In this case it is known that W_{1} is the smooth locus of Ω ([13, Theorem 5.3]). Since PGL(H_{0}) acts smoothly on H_{0}, any f ∈ Aut(Ω) must preserve W_{1}.

Proof. We observe that any f ∈ Aut(Ω) preserves the set of maximal linear subspaces of Ω (if π is maximal and π′ ⊋ f(π) then f^{−1}(π′) ⊋ π). We consider three cases: m = 2ℓ − 1, m = 2ℓ + 1 and m ≠ 2ℓ ± 1. If m = 2ℓ − 1, then it follows from Lemma 5.1, that the only maximal linear subspaces of Ω of dimension ℓ − 2 are the π̃_{β}’s. (Since m ≥ 3, we have ℓ − 2 ≥ 0, therefore π̃_{β} has at least one point). Therefore the set of π̃_{β}’s is preserved, and hence f(W_{1}) = W_{1} by Lemma 5.4. If m = 2ℓ+1 then, it follows from Lemma 5.1, that the only maximal linear subspaces of Ω of dimension ℓ − 1 are the π̃^{δ}’s. Therefore the set of π̃^{δ}’s is preserved, and hence f(W_{1}) = W_{1} by Lemma 5.4. Finally if m ≠ 2ℓ ± 1, then the only maximal linear subspaces of dimension m − ℓ − 1 are the π̃_{β}’s (and also the π̃^{δ}’s, if m = 2ℓ).

Thus f preserves the set of π̃_{β}’s (or the π̃_{β}’s and π̃^{δ}’s taken together if m = 2ℓ). By Lemma 5.4, f(W_{1}) = W_{1}. □

For each γ ∈ W_{0}, the decomposition V = V_{m−ℓ} ⊕ γ induces a natural inclusion ψ̂ : ∧^{1}V_{m−ℓ} ⊗ ∧^{ℓ−1}γ ↪ ∧^{ℓ}V that takes a decomposable element v ⊗ β to v ∧ β. Let ψ : ℙ(V_{m−ℓ} ⊗∧^{ℓ−1}γ) ↪ ℙ(∧^{ℓ}V) denote the corresponding projective map. We note that ψ bijectively carries decomposable elements v ⊗ β to decomposable elements v ∧ β ∈ W_{1} ⊂ Ω. Since Ω ⊂ H_{0}, we see im(ψ) is an ℓ(m − ℓ) − 1-dimensional linear subspace of H_{0}. The decomposable elements of ℙ(V_{m−ℓ} ⊗ ∧^{ℓ−1}γ) can be identified with ℙ(V_{m−ℓ}) × ℙ(∧^{ℓ−1}γ) by the Segre embedding. We define:

(18)  Δ_{γ} := ψ(ℙ(V_{m−ℓ}) × ℙ(∧^{ℓ−1}γ)) = im(ψ) ∩ Ω.

Thus the subvariety Δ_{γ} ⊂ Ω is an embedding of ℙ^{m−ℓ−1} × ℙ^{ℓ−1} into Ω. For each v ∈ ℙ(V_{m−ℓ}), ψ(v ⊗ ∧^{ℓ−1}γ) is just π̃^{δ} where δ ∈ G_{ℓ+1}(V) is represented by v ∧ γ. For each β ∈ ℙ(∧^{ℓ−1}γ) = G_{ℓ−1}(γ), we have ψ(ℙ(V_{m−ℓ}) ⊗ β) is just π̃_{β}. Therefore, Δ_{γ} can also be described as a union of maximal linear subspaces of W_{1}:

(19)  Δ_{γ} = ⋃_{β⊂γ} π̃_{β} = ⋃_{δ⊃γ} π̃^{δ}

For γ ∈ W_{0} and β ≠ β′ ∈ G_{ℓ−1}(γ) and δ ≠ δ′ ∈ G_{ℓ+1}(V) containing γ, we have β + β′ = δ ∩ δ′ = γ ∉ W_{1}. Therefore it follows by Lemma 5.2 that the unions appearing in (19) are disjoint.

The next lemma and proposition concern properties of Δ_{γ} which are needed in Theorem 5.9.

Lemma 5.6. Let γ, γ′ ∈ W_{0}. Then:

(i) Each line in G_{ℓ}(V) through γ intersects Ω in a unique point. The set of all such intersection points is precisely Δ_{γ}.

(ii) Δ_{γ} = Δ_{γ′} if and only if γ = γ′.

Proof. (i) A line π_{β}^{δ} in G_{ℓ}(V) passes through γ ∈ W_{0} if and only if β ∈ W_{0}^{−} and δ ∈ W_{0}^{+} with β ⊂ γ ⊂ δ. The intersection of π_{β}^{δ} ∩ Ω equals π̃_{β} ∩ π̃^{δ} = β ∧ v_{δ} where v_{δ} ∈ ℙ(V_{m−ℓ}) is the point ℙ(δ ∩ V_{m−ℓ}), as noted in Lemma 5.2. Therefore, the set of intersection points of lines through γ with Ω is

⋃_{β⊂γ} π̃_{β}

and this union is Δ_{γ}, thanks to (19).

(ii) Suppose Δ_{γ} = Δ_{γ′}. For any β′ ∈ G_{ℓ−1}(γ′), we have π̃_{β′} ⊂ Δ_{γ′}, hence π̃_{β′} ⊂ Δ_{γ}. Since Δ_{γ} is the disjoint union of π̃^{γ∧v} as v runs over ℙ(V_{m−ℓ}), we must have β′ ∧ v ∈ π̃^{γ∧v}, because v is the unique element of ℙ(V_{m−ℓ} ∩ (β ∧ v)) as well as ℙ(V_{m−ℓ} ∩ (γ ∧ v)). Therefore, β′ ⊂ (γ ⊕ 𝕂v̂) for all v̂ ∈ V_{m−ℓ}\{0}. In particular, for two linearly independent vectors v̂, ŵ ∈ V_{m−ℓ}, we see β′ ⊂ (γ⊕𝕂v̂)∩(γ⊕𝕂ŵ) which is γ (because γ ∩ V_{m−ℓ} = {0}). We have thus shown that every (ℓ − 1)-dimensional subspace of γ′ is also a subspace of γ, which implies γ = γ′. Conversely, if γ = γ′, then clearly Δ_{γ} = Δ_{γ′}. □

Proposition 5.7. Let X, Y and Z be 𝕂-vector spaces of dimensions m − ℓ, ℓ and (m − ℓ)ℓ respectively such that ℙ(Z) a linear subspace of the hyperplane H_{0}. Let Ψ : ℙ(X⊗Y) → ℙ(Z) be a projective semilinear isomorphism carrying decomposable elements (i.e., ℙ(X) × ℙ(Y), by Segre embedding) to decomposable elements (i.e., ℙ(Z)∩Ω ). Suppose that on ℙ(X)×ℙ(Y), Ψ takes [x]×ℙ(Y) to a π̃^{δ(x)} and ℙ(X)×[y] to a π̃_{β(y)}, for each [x] ∈ ℙ(X) and [y] ∈ ℙ(Y). Then Ψ(ℙ(X) × ℙ(Y)) = Δ_{γ} for some γ ∈ W_{0}.

Proof. Let [x], [y], δ(x) and β(y) be as in the hypothesis. Let Ψ_{1}(x) denote the point ℙ(δ(x)∩V_{m−ℓ}) of ℙ(V_{m−ℓ}). Now, any element γ′ ⊂ π̃_{β(y)} can be written uniquely as v ∧ β(y) where v is the point ℙ(γ′ ∩ V_{m−ℓ}) of ℙ(V_{m−ℓ}). Let γ′ := Ψ([x ⊗ y]) ∈ π̃^{δ(x)}. Then Ψ_{1}(x) is ℙ(γ′ ∩ V_{m−ℓ}), hence Ψ([x ⊗ y]) = Ψ_{1}(x) ∧ β(y). Since Ψ_{|(ℙ(X)×[y])} : ℙ(X) → π̃_{β(y)} is a bijection, we conclude that Ψ_{1} : ℙ(X) → ℙ(V_{m−ℓ}) is a bijection. Let [y] ≠ [y′], then ℙ(X) × [y] and ℙ(X) × [y′] are disjoint, and since Ψ is injective we get π̃_{β(y)} and π̃_{β(y′)} are disjoint. In particular, we get β(y) ≠ β(y′) and hence ℓ ≤ dim(β(y)+β(y′)). Since Ψ_{1}(x)∧β(y), Ψ_{1}(x)∧β(y′) ∈ π̃^{δ(x)} for each [x] ∈ ℙ(X), both β(y) and β(y′) are contained in δ(x), and hence dim(β(y) + β(y′)) ≤ ℓ + 1. In case this dimension is ℓ + 1, it follows that β(y) + β(y′) = δ(x) for all [x] ∈ ℙ(X). In particular Ψ_{1}(x) = ℙ(δ(x) ∩ V_{m−ℓ}) is independent of [x] contradicting the fact proved above that Ψ_{1} is bijective. Therefore, γ := β(y) + β(y′) is ℓ-dimensional, and the fact that dim(β(y) ∩ V_{m−ℓ}) = 0 implies dim(γ ∩ V_{m−ℓ}) ≤ 1, i.e γ ∈ W_{0} or W_{1}. Since π̃_{β(y)} and π̃_{β(y′)} are disjoint, it follows by Lemma 5.2 that γ ∈ W_{0}. Writing δ(x) = γ ∧ Ψ_{1}(x) for each [x] ∈ ℙ(X), we see that {δ(x) : x ∈ ℙ(X)} is the family of all δ ∈ G_{ℓ+1}(V) containing γ ∈ W_{0}. Therefore:

Ψ(ℙ(X) × ℙ(Y)) = ⋃_{x∈ℙ(X)} Ψ([x] × ℙ(Y)) = ⋃_{x∈ℙ(X)} π̃^{δ(x)} = ⋃_{δ⊃γ} π̃^{δ}

which by (19) equals Δ_{γ}. □

The next lemma is needed for the case m = 2ℓ in Theorem 5.9.

Lemma 5.8. Let γ ∈ W_{0}, β ∈ W_{0}^{−}, and δ ≠ δ′ ∈ W_{0}^{+}

(i) If π̃_{β} and π̃^{δ} intersect, then the only lines in Ω joining a point of π̃_{β} to a point of π̃^{δ} are the ones passing through the point π̃_{β} ∩ π̃^{δ} and completely contained in π̃_{β} or π̃^{δ}.

(ii) If π̃^{δ} and π̃^{δ′} intersect, then there is a line in Ω joining a point of π̃^{δ} to a point of π̃^{δ′} which is not completely contained in W_{1}.

Proof. (i) Let 𝕂v_{0} = V_{m−ℓ} ∩ δ. As observed in the discussion after Lemma 5.1, π̃_{β} ∩ π̃^{δ} = 𝕂v_{0} ⊕ β. Now, let L be a line joining a point γ_{1} ∈ π̃_{β} to some point γ_{2} ∈ π̃^{δ}. Let 𝕂v_{1} = V_{m−ℓ} ∩ γ_{1}. If v_{0} and v_{1} are dependent, then clearly γ_{1} = 𝕂v_{0} ⊕ β = π̃_{β} ∩ π̃^{δ}, and therefore L ⊂ π̃^{δ}. If v_{0} and v_{1} are independent, then we pick a complement γ_{0} of 𝕂v_{0} in δ satisfying β ⊂ γ_{0}. Every element of π̃^{δ} (and hence γ_{2} in particular) is of the form 𝕂v_{0} ⊕ β′ for some (ℓ − 1)-dimensional subspace β′ of γ_{0}. We write γ_{1} + γ_{2} = (𝕂v_{0} ⊕ 𝕂v_{1}) ⊕ (β + β′) (where we have used the fact that (β + β′) ∩ V_{m−ℓ} ⊂ γ_{0} ∩ V_{m−ℓ} = {0}). By definition of a line in G_{ℓ}(V), dim(γ_{1} + γ_{2}) = ℓ + 1, and hence dim(β + β′) = ℓ − 1, which implies β = β′. Thus γ_{2} ∈ π̃_{β} and hence L ⊂ π̃_{β}.

(ii) By definition of π̃^{δ}, it follows that π̃^{δ} and π̃^{δ′} intersect if and only if δ∩V_{m−ℓ} = δ′ ∩ V_{m−ℓ} = 𝕂v_{0} (for some v_{0} ∈ V_{m−ℓ}), and γ := δ ∩ δ′ ∈ W_{1} ⊂ G_{ℓ}(V). We write δ = γ ⊕𝕂u and δ′ = γ ⊕𝕂u′. Since δ +δ′ is (ℓ+2)-dimensional there is a v_{1} ∈ V_{m−ℓ} contained in δ + δ′ and independent of v_{0}. We write v_{1} = u′′ + au + bu′ for some u′′ ∈ γ and a, b ∈ 𝕂. Since v_{1} ∉ δ, δ′, the scalars a, b are both non-zero. Hence we can express δ = γ ⊕ 𝕂u′′′ where u′′′ = u′′ + au. Now for any (ℓ − 1)-dimensional subspace β of γ containing v_{0}, we observe that the line joining β ⊕ 𝕂u′′′ ∈ π̃^{δ} and β ⊕ 𝕂u′ ∈ π̃^{δ′} is not contained in W_{1} as it contains β ⊕ 𝕂v_{1} ∈ W_{2}. □

Theorem 5.9. The automorphism group of the Schubert divisor Ω is given by

Aut(Ω) ≃ Aut(W_{0}) = \begin{cases}ρ(P̂_{m−ℓ,ℓ}/𝕂^{×}) & if m ≠ 2ℓ, \\ ρ(P̂_{ℓ,ℓ}/𝕂^{×}) ⋊_{˜∗_{ℓ}} ℤ/2ℤ & if m = 2ℓ. \end{cases}

More precisely, for each f ∈ Aut(Ω), there is a unique way to extend f to an automorphism ι(f) of G_{ℓ}(V). The image of this monomorphism from Aut(Ω) to Aut(G_{ℓ}(V)) is the subgroup Aut(W_{0}) of Aut(G_{ℓ}(V)).

Proof. By Theorem 4.2, Aut(W_{0}) is the subgroup of Aut(G_{ℓ}(V)) that preserves W_{0}, and hence its complement Ω. Every element of Aut(W_{0}) preserves H_{0}: Writing f ∈ Aut(W_{0}) as ∧^{ℓ}g (or also (∧^{ℓ}g)˜∗_{ℓ} if m = 2ℓ) where ĝ(x) = [ \begin{matrix}A & u \\ 0 & B \end{matrix}] μ(x) is a representative of g in ΓL(V), we see that p_{I_{0}}(∧^{ℓ}ĝ(ξ)) = μ(p_{I_{0}}(ξ))det(B). Therefore ∧^{ℓ}g preserves H_{0}. In case m = 2ℓ, it was shown in the discussion following Definition 4.3 that ˜∗_{ℓ} ∈ Aut(W_{0}) preserves H_{0}. Therefore, we have a restriction homomorphism:

res_{Ω} : Aut(W_{0}) → Aut(Ω) given by f ↦ f_{|H_{0}}.

We claim that this homomorphism is injective, i.e., if f ∈ Aut(W_{0}) fixes Ω pointwise, then it fixes W_{0} pointwise. We will show f(γ) = γ for each γ ∈ W_{0}. By Lemma 5.6, the set of intersection points of all lines through γ ∈ W_{0} with Ω is Δ_{γ}. Since f is also an automorphism of G_{ℓ}(V) (by Theorem 4.2), these lines get mapped to the set of all lines through f(γ) ∈ W_{0}. Since f fixes Ω pointwise, we see that the intersection points with Ω of all lines through f(γ) is Δ_{γ}, however it is also Δ_{f(γ)} by Lemma 5.6. Thus, we get Δ_{γ} = Δ_{f(γ)}, and part (ii) of Lemma 5.6 implies γ = f(γ).

In order to show res_{Ω} is surjective, we will construct an extension of each f ∈ Aut(Ω) to an automorphism ι(f) of G_{ℓ}(V) that preserves W_{0}, i.e., such that res_{Ω}(ι(f)) = f for all f ∈ Aut(Ω). Consequently, res_{Ω} is an isomorphism and the function f ↦ ι(f) will be the inverse isomorphism to res_{Ω}. By Proposition 5.5, each f ∈ Aut(Ω) satisfies f(W_{1}) = W_{1}, and hence f preserves the set of maximal linear subspaces of W_{1}. First we assume that no f(π̃_{β}) is a π̃^{δ} (this is automatic if m ≠ 2ℓ, by looking at their dimensions). Thus we have bijections f_{−} : W_{0}^{−} → W_{0}^{−} and f_{+} : W_{0}^{+} → W_{0}^{+} defined by f(π̃_{β}) = π̃_{f_{−}(β)} and f(π̃^{δ}) = π̃^{f_{+}(δ)}. We now fix a γ ∈ W_{0}, and recall the linear isomorphism ψ : ℙ(V_{m−ℓ} ⊗ ∧^{ℓ−1}γ) → im(ψ) ⊂ H_{0} (defined after Proposition 5.5). Let X = V_{m−ℓ}, Y = ∧^{ℓ−1}γ, and let Z be defined by ℙ(Z) = im(f ∘ ψ). Since im(ψ) ⊂ H_{0} and f is a semilinear isomorphism of H_{0}, the composition Ψ := f ∘ ψ : ℙ(X ⊗ Y) ↪ ℙ(Z) ⊂ H_{0}, is a projective semilinear isomorphism, satisfying Ψ(v ⊗ ∧^{ℓ−1}γ) = π̃^{f_{+}(δ)} (where δ = v ∧ γ), and Ψ(V_{m−ℓ} ⊗ β̂) = π̃_{f_{−}(β)} for each v ∈ ℙ(V_{m−ℓ}) and β ∈ ℙ(∧^{ℓ−1}γ). Therefore, Ψ satisfies all the hypothesis of Proposition 5.7, and hence Ψ(ℙ(X) × ℙ(Y)) = f(Δ_{γ}) is a Δ_{f_{0}(γ)} for some f_{0}(γ) ∈ W_{0}. This defines a function f_{0} : W_{0} → W_{0} which has an inverse namely the corresponding function (f^{−1})_{0}. Thus f_{0} is a bijection. We now have a bijection ι(f) : G_{ℓ}(V) → G_{ℓ}(V) defined by the pair f, f_{0}. In order to show ι(f) ∈ Aut(G_{ℓ}(V)) we must show that ι(f) and ι(f^{−1}) carry lines to lines.

Let L be a line in G_{ℓ}(V). If L does not intersect W_{0}, then L ⊂ Ω and hence f(L) and f^{−1}(L) are lines in Ω. If L passes through a point γ ∈ W_{0}, then writing L = π_{β}^{δ} we must have β ⊂ γ ⊂ δ with dim(δ ∩V_{m−ℓ}) = 1. Now, except the point π̃_{β}^{δ} := π̃_{β} ∩ π̃^{δ}, the remaining points of π_{β}^{δ} are all in W_{0}. For each γ′ ∈ π_{β}^{δ} ∩ W_{0}, it follows from the decomposition (19) of Δ_{γ′} that π̃_{β}, π̃^{δ} ⊂ Δ_{γ′}. Hence π̃_{f_{−}(β)}, π̃^{f_{+}(δ)} ⊂ Δ_{f_{0}(γ′)}, thus proving that f_{−}(β) ⊂ f_{0}(γ′) ⊂ f_{+}(δ), i.e., f_{0}(π_{β}^{δ} ∩ W_{0}) ⊂ π_{f_{−}(β)}^{f_{+}(δ)}. Moreover

f carries the remaining point of π_{β}^{δ}, namely π̃_{β} ∩ π̃^{δ} to π̃_{f_{−}(β)} ∩ π̃^{f_{+}(δ)} ∈ π_{f_{−}(β)}^{f_{+}(δ)}.

Thus ι(f) takes the line π_{β}^{δ} to the line π_{f_{−}(β)}^{f_{+}(δ)}. Repeating this argument for f^{−1}, we conclude that ι(f), ι(f^{−1}) are line preserving bijections of G_{ℓ}(V) (which preserve W_{0}), hence ι(f) ∈ Aut(W_{0}).

It remains to consider the case when m = 2ℓ and there exists a β_{0} with f(π̃_{β_{0}}) = π̃^{δ_{0}}. In this case, for each γ ∈ W_{0} with β_{0} ⊂ γ we will prove that {f(π̃_{β}) : β ⊂ γ} are all π̃^{δ}’s, and {f(π̃^{δ}) : δ ⊃ γ} are all π̃_{β}’s. Using a connectivity argument (as in the proof of Chow’s theorem) we will then show that f interchanges the sets {π̃_{β} : β ∈ W_{0}^{−}} and {π̃^{δ} : δ ∈ W_{1}^{+}}. To begin with, suppose f(π̃_{β_{0}}) = π̃^{δ_{0}}. Let v_{0} = ℙ(δ_{0} ∩ V_{m−ℓ}), and pick an arbitrary γ ∈ W_{0} containing β_{0}. For any β′ ⊂ γ with β′ ≠ β_{0}, suppose f(π̃_{β′}) = π̃_{β′′}. Since f(π̃_{β′}) = π̃_{β′′}, there is a v′_{0} ∈ ℙ(V_{m−ℓ}), such that f(v′_{0} ∧ β′) = v_{0} ∧ β′′. We consider π̃^{δ′} where δ′ = v′_{0} ∧ γ. Let L be the line in π̃^{δ′} joining v′_{0} ∧ β′ and v′_{0} ∧ β_{0}. The line f(L) joins v_{0} ∧ β′′ to some point v_{0} ∧ β′′′ ∈ π̃^{δ_{0}}. In particular every γ′′ ∈ f(L) contains v_{0} (because each γ′′ ⊃ (v_{0} ∧ β′′) ∩ (v_{0} ∧ β′′′)). Since the intersection of all γ′′ lying on a line in a π̃_{β} is β, and ℙ(β ∩ V_{m−ℓ}) = ∅, we see that f(π̃^{δ′}) must be a π̃^{δ′′} for some δ′′ ∈. Since v_{0} ∧ β′′′ is a common point of π̃^{δ′′} and π̃^{δ_{0}}, part (ii) of Lemma 5.8 implies that there is a line joining a point of π̃^{δ′′} to a point of π̃^{δ_{0}} which is not contained in W_{1}. However every such line is the image under f of a line joining a point of π̃^{δ′} to a point of π̃_{β_{0}}, which by part (i) of Lemma 5.8 is contained in W_{1}. This contradicts the fact that f preserves W_{1}, and therefore we conclude that f carries all π̃_{β}’s in Δ_{γ} to π̃^{δ}’s. Now, suppose there is a π̃^{δ} in Δ_{γ} such that f(π̃^{δ}) is a π̃^{δ′}. Now π̃_{β_{0}} and π̃^{δ} intersect in a point, and hence their images under f, namely π̃^{δ_{0}} and π̃^{δ′} also intersect in a point. Again using parts (ii) and (i) of Lemma 5.8, there is a line joining a point of π̃^{δ_{0}} to a point of π̃^{δ′} which is not contained in W_{1}, but it is the image under f of a line contained in W_{1}, which contradicts the fact that f preserves W_{1}, and therefore we conclude that f carries all π̃^{δ}’s in Δ_{γ} to π̃_{β}’s. In the reasoning above, γ ∈ W_{0} containing β_{0} was arbitrary. Therefore, in order to show that f interchanges the sets {π̃_{β} : β ∈ W_{0}^{−}} and {π̃^{δ} : δ ∈ W_{1}^{+}}, it suffices to show that given γ, γ′′ ∈ W_{0} there is a sequence γ = γ_{0}, γ_{1}, …, γ_{ℓ} = γ′′ such that γ_{i−1} and γ_{i} intersect in a β_{i} ∈ W_{0}^{−}. Now, every γ ∈ W_{0} has a unique expression of the form γ = (e_{m−ℓ+1} + v_{1}) ∧ ⋯ ∧ (e_{m} + v_{ℓ}) where v_{1}, …, v_{ℓ} ∈ V_{m−ℓ}. Given Δ_{γ′′} with γ′′ = (e_{m−ℓ+1} + v′_{1}) ∧ ⋯ ∧ (e_{m} + v′_{ℓ}), let γ_{0} = γ, γ_{ℓ} = γ′′, then the desired sequence is γ_{i} := (e_{m−ℓ+1} + v′_{1}) ∧ ⋯ ∧ (e_{m−ℓ+i} + v′_{i}) ∧ (e_{m−ℓ+i+1} + v_{i+1}) ∧ ⋯ ∧ (e_{m} + v_{ℓ}).

Now ˜∗_{ℓ} ∈ Aut(G_{ℓ}(V)) interchanges the set of π_{β}’s and the set of π^{δ}’s. Since ˜∗_{ℓ} preserves Ω and π̃_{β} = π_{β} ∩ Ω, π̃^{δ} = π^{δ} ∩ Ω (as observed in the proof of Lemma 5.1) we conclude that ˜∗_{ℓ} ∈ Aut(Ω) interchanges the set of π̃_{β}’s and the set of π̃^{δ}’s. If f ∈ Aut(Ω) interchanges the set of π̃_{β}’s and the set of π̃^{δ}’s, then ˜∗_{ℓ} ∘ f preserves the set of π̃_{β}’s as well as the set of π̃^{δ}’s, and therefore ˜∗_{ℓ}^{−1} ∘ ι(˜∗_{ℓ} ∘ f) is the desired extension ι(f) of f to Aut(W_{0}). This concludes the construction of the isomorphism f ↦ ι(f) between Aut(Ω) and Aut(W_{0}). □

Remark: Mark Pankov has brought to our attention the work of Prażmowski and Żynel [19] which shows that automorphisms of certain subspaces of G_{ℓ}(V) called spine spaces (see [17, Section 3.6] for a definition) extend to automorphisms of G_{ℓ}(V). The spaces W_{i} defined in (15) are examples of spine spaces.

Corollary 5.10. For the Schubert divisor code C_{Ω}(ℓ, m), we have

Aut(C_{Ω}(ℓ, m))/F^{×} ≃ \begin{cases}ρ(P̂_{m−ℓ,ℓ}/F^{×}) & if m ≠ 2ℓ, \\ ρ(P̂_{ℓ,ℓ}/F^{×}) ⋊_{˜∗_{ℓ}} ℤ/2ℤ & if m = 2ℓ. \end{cases}

The groups Aut(C_{Ω}(ℓ, m)) and MAut(C_{Ω}(ℓ, m)) are isomorphic to Aut(C^{𝔸}(ℓ, m)) and MAut(C^{𝔸}(ℓ, m)), respectively, and hence they can be explicitly described as in Theorem 4.5.

Proof. Since Aut(C_{Ω}(ℓ, m))/F^{×} is just Aut(Ω), the first assertion of the corollary follows from Theorem 5.9. Let Ĥ_{0} be the hyperplane in ∧^{ℓ}V given by the vanishing of the Plücker coordinate p_{I_{0}} where I_{0} := (m − ℓ + 1, m − ℓ + 2, …, m). Let p_{2} : ΓL(Ĥ_{0}) → PΓL(H_{0}) and p_{1} : ΓL(∧^{ℓ}V) → PΓL(∧^{ℓ}V) be the projection homomorphisms. The group Aut(C_{Ω}(ℓ, m)) is isomorphic to p_{2}^{−1}(Aut(Ω)), whereas the group Aut(C^{𝔸}(ℓ, m)) is isomorphic to p_{1}^{−1}(Aut(W_{0})). Given any f ∈ Aut(Ω), let f̂ ∈ p_{1}^{−1}(ι(f)). Since ι(f) preserves H_{0}, we get a homomorphism ȷ : Aut(C^{𝔸}(ℓ, m)) → Aut(C_{Ω}(ℓ, m)) obtained by restricting each f̂ to Ĥ_{0}. Now any g ∈ p_{2}^{−1}(f) is of the form c ȷ(f̂) for some non-zero scalar c. Since c ȷ(f̂) = ȷ(c f̂), it follows that ȷ is surjective. Suppose f̂ ∈ ker(ȷ), then f = p_{2} ∘ ȷ(f̂) is the identity of Aut(Ω), and hence ι(f) is the identity of Aut(W_{0}), whence f̂ is c times the identity of ΓL(∧^{ℓ}V). Therefore, ȷ(f̂) is c times the identity of ΓL(Ĥ_{0}) which implies c = 1, i.e., ȷ is injective. Thus the homomorphism ȷ is an isomorphism between Aut(C_{Ω}(ℓ, m)) and Aut(C^{𝔸}(ℓ, m)). Moreover, since the isomorphism ȷ simply maps elements of Aut(C^{𝔸}(ℓ, m)) to their restrictions to the hyperplane Ĥ_{0}, it follows that MAut(C_{Ω}(ℓ, m)) is isomorphic to MAut(C^{𝔸}(ℓ, m)) as well. □

We remark that Aut(C(ℓ, m)) is transitive in the sense that given any pair e_{i} ≠ e_{j} of basic vectors of F^{n}, there is an element of Aut(C(ℓ, m))which sends e_{i} to a scalar multiple of e_{j}. In contrast, Aut(C_{Ω}(ℓ, m)) is not transitive since it preserves the basic vectors representing W_{i}, for each i = 1, …, ℓ. As remarked before, in the case when the field F is replaced by an algebraically closed field, this lack of transitivity is reflected in the fact that Ω is not a smooth variety, and the automorphisms must preserve the smooth locus W_{1} ⊂ Ω.

Acknowledgement

We are grateful to Mark Pankov for his comments on an initial version of this article and for bringing [17], [22] and [19] to our attention. Krishna Kaipa would like to thank the Department of Mathematics at IIT Bombay where this work was carried out.

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Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

E-mail address: srg@math.iitb.ac.in

Department of Mathematics, Indian Institute of Science Education and Research-Bhopal, Bhauri, Bhopal 462030, India

E-mail address: kaipa@iiserb.ac.in