On the Inversion and Dimension Pairs of Row-Strict Tableaux

Abstract

In this article, we consider two algorithms, dimension and inversion pairs of rows-strict, used for the computation of Betti numbers of Springer varieties and then show that the sequences respectively generated by these algorithms are dual to each other, (except for λ = 1ⁿ where I_k = D_k) and that the sum I_k + D_k gives another sequence which is palindromic. We also show that for each row-strict tableau τ of shape λ = n − r, 1^r (0 ≤ r ≤ n − 1), the dimension of the corresponding Springer varieties equals the cardinality of the union of the set of inversions and dimensions of τ. This research contributes to a deeper understanding of the rich combinatorial landscape of tableaux, opening up new avenues for further research.