Brundan-kazhdan-lusztig and Super Duality Conjectures

Perverse Sheaves and the Kazhdan-lusztig Conjectures

Proof of Two Conjectures of Brenti and Simion on Kazhdan-Lusztig Polynomials

We find an explicit formula for the Kazhdan-Lusztig polynomials Pui,a ,vi of the symmetric group S(n) where, for a, i, n ∈ N such that 1 ≤ a ≤ i ≤ n, we denote by ui,a = sasa+1 · · · si−1 and by vi the element ofS(n) obtained by inserting n in position i in any permutation ofS(n −1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two c...

Quantum Schur Superalgebras and Kazhdan–lusztig Combinatorics

We introduce the notion of quantum Schur (or q-Schur) superalgebras. These algebras share certain nice properties with q-Schur algebras such as base change property, existence of canonical Z[v, v]-bases, and the duality relation with quantum matrix superalgebra A(m|n). We also construct a cellular Q(υ)-basis and determine its associated cells, called super-cells, in terms of a Robinson–Schenste...

A Flag Whitney Number Formula for Matroid Kazhdan-Lusztig Polynomials

For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes the intersection cohomology of the associated reciprocal plane. However, these polynomials are difficult to compute and there are numerous open conjectures about their structure. For example, it is unknown whether or not the coefficients are non-negative for non-representable matroids. The main res...

The Z-Polynomial of a Matroid

We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. For realizable matroids, we give a cohomologica...