Mathematics Colloquium-II
Date and time: Thursday, 1 December 2022, 3 pm
Venue: Ramanujan Hall
Speaker: Prashant Singh, IIT Jammu
Title: Maximal Hyperplane Sections of Determinantal Varieties of Symmetric Matrices over Finite Fields
Abstract: Let X = (Xij ) be an m × n generic matrix whose entries are independent indeterminates over a field F. The classical determinantal variety Dt = Dt(m,n) given by the vanishing of (t+1)×(t+1) minors of X has been extensively studied since antiquity. Since the defining equations have coefficients ±1, the variety is also defined over any finite field Fq. In fact, Dt has a large number of Fq-rational points and that makes it a useful object from the point of view of applications to coding theory. An explicit formula for |Dt(Fq)| is well-known and goes back at least to Landsberg (1893). Partly from the viewpoint of applications, one is also interested in the following questions concerning the cardinalities of hyperplane sections of Dt:
(i) What are the possible values of |Dt∩H(Fq)|, where H is a F_q-rational hyperplane in the projective space?
(ii) What is the maximum possible value of |Dt ∩H(Fq)|, where H varies over the hyperplanes as in (i) above?
These have been answered relatively recently. In fact, various approaches for the first question are possible, e.g., using the eigenvalues of certain association schemes or variants of the so-called MacWilliams identities. We will begin by reviewing these developments. Then we consider the case where X = (Xij) is a generic symmetric matrix of size m × m, and look at the corresponding variety St = St(m) given by the vanishing of (t + 1) × (t + 1) minors of X. Here, the number of Fq-rational points were determined by Carlitz (1954) in a special case, and by MacWilliams (1969) in the general case. In this case, the question analogous to (i) is open, in general, whereas an answer to (ii) has been obtained recently when t is even, while a conjectural answer is proposed when t is odd. We will give a motivated account of these results, which are obtained in joint work with Peter Beelen and Trygve Johnsen.