Abstract
A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Finally, we share some the results of a computer program developed to calculate tableaux inversions.
Highlights
Let λ = (λ1, λ2, . . . , λm) be a non-increasing sequence of positive integers that partition N
Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of i-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber
We develop formulas enumerating the number of i-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableau that standardize a specific standard Young tableau, and construct bijections between i-inverted Young tableaux of a certain shape with j-inverted Young tableaux of different shapes
Summary
In [3], Fresse utilizes inverted Young tableaux to determine the Betti numbers of Springer fibers in type A He fixes a specific standard Young tableau T of shape λ and considers the corresponding set of inverted tableaux inv(T ) = {τ | st(τ ) = T }. He argues that the component of the Springer variety Fλ corresponding to T has mth Betti number equaling the number of inverted Young tableaux in inv(T ) with precisely d − m inversions, where d is the dimension of the entire Springer variety. We avoid this difficulty by presenting techniques for directly calculating the sizes of the entire sets Si(λ)
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