Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
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支持斜杨图和罗特图补集的有限域上的矩阵计数
DOI：
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发表时间：
2012
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通讯作者：
A. Morales
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文献类型：
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作者：
Aaron J. Klein;J. Lewis;A. Morales
我们考虑了在有限域上求矩阵个数的问题，该有限域具有一定的秩且支撑度避免了一个子集的条目。这些矩阵是具有受限位置(即，ROK放置)的排列的Q-模拟。对于一般的条目集，这些矩阵的数目不是Q(在ANN中为Stembridge)中的多项式。梳子。2(4)：365,1998年)；然而，当项目集是杨图时，数字，高达Q−1的幂，是具有非负系数的多项式(Haglund在Adv.Appl.数学课。20(4)：450,1998)。在本文中，我们给出了这些数是Q中的多项式，甚至是具有非负整系数的多项式的若干条件。我们将Haglund的结果推广到斜Young图的补图上，并将这一结果应用于元素集是置换的Rothe图的情况。特别地，我们给出了关于其Rothe图是斜Young图的补图直至行和列重新排列的置换的充要条件。最后，我们给出了一些猜想，其中可逆矩阵的支撑避免了Rothe图和强Bruhat阶的Poincaré多项式。
We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries, these numbers of matrices are not polynomials in q (Stembridge in Ann. Comb. 2(4):365, 1998); however, when the set of entries is a Young diagram, the numbers, up to a power of q−1, are polynomials with nonnegative coefficients (Haglund in Adv. Appl. Math. 20(4):450, 1998).In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund’s result to complements of skew Young diagrams, and we apply this result to the case where the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincaré polynomials of the strong Bruhat order.