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864.05087
Ghorpade, Sudhir R.
Young multitableaux and higher dimensional determinants. (English)
[J] Adv. Math. 121, No.2, 167-195, Art. No.0051 (1996). [ISSN 0001-8708]
A standard Young multitableau of width $q>0$ bounded by $m=(m(1),\dots,m(q))$ is a $q$-tuple of standard Young tableaux of the same shape such that the entries of the $i$th tableau are bounded by $m(i)$. Let $\text{stab}(q,m,p,a,V)$ be the number of the standard multitableaux $T$ of width $q$, with $V$ boxes, bounded by $m$ and predominated by the multivector $a$ of width $q$ and length $p$ (i.e. placing $a$ before the first row of $T$, the resulting tableau is standard again). The main result of the paper under review is a formula which, fixing $q$ even, $p$, $m$ and $a$, gives a polynomial in $V$ which is equal to the cardinality of $\text{stab}(q,m,p,a,V)$. The main motivation for this result comes from the work of Abhyankar on enumerating Young multitableaux. It also generalizes the important Abhyankar formula for $\text{stab}(2,m,p,a,V)$ which is a basic ingredient of Abhyankar's enumerative proof of the straightening law of Doubilet-Rota-Stein. Using the notion of a $q$-dimensional determinant, as a consequence of the main result of this paper, the author shows that the natural generalization of the straightening law for multitableaux is not true in general for even $q>2$. The author also answers several problems posed in [{\it S. S. Abhyankar}, Enumerative combinatorics of Young tableaux, Dekker, New York-Basel (1988; Zbl 643.05001)].
[ V.Drensky (Sofia) ]
- MSC 1991:
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*05E10 Tableaux, etc.
15A15 Special matrix functions
05A15 Combinatorial enumeration problems
05A10 Combinatorial functions
Keywords: Young multitableau; Young tableaux; Abhyankar formula; straightening law
Citations: Zbl 643.05001
Cited in Zbl. reviews...
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