Gregory S. Warrington

Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

AbstractIn (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials Px,w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where

Counterexamples to the 0-1 Conjecture

W = \mathfrak{S}_n

Quasisymmetric expansions of Schur-function plethysms

(the symmetric group on n letters) and the permutation w is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for w. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincaré polynomial of the intersection cohomology of the Schubert variety corresponding to w is (1+q)l(w) if and only if w is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety Xw to have a small resolution. We conclude with a simple method for completely determining the singular locus of Xw when w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (BCn, F4, G2).

What to Expect in a Game of Memory

Schubert varieties in the flag manifold SL(n)/B play a key role in our understanding of projective varieties. One important problem is to determine the locus of singular points in a variety. In 1990, Lakshmibai and Sandhya showed that the Schubert variety X w is nonsingular if and only if w avoids the patterns 4231 and 3412. They also gave a conjectural description of the singular locus of X w . In 1999, Gasharov proved one direction of their conjecture. In this paper we give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety X w for any element w ∈ G n . In doing so, we prove both directions of the Lakshmibai-Sandhya conjecture. These irreducible components are indexed by permutations which differ from w by a cycle depending naturally on a 4231 or 3412 pattern in w. Our description of the irreducible components is computationally more efficient (O(n 6 )) than the previously best known algorithms, which were all exponential in time. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.

Equivalence Classes for the μ-Coefficient of Kazhdan–Lusztig Polynomials in Sn

For permutations x and w, let μ(x, w) be the coefficient of highest possible degree in the Kazhdan-Lusztig polynomial Px,w. It is well-known that the μ(x, w) arise as the edge labels of certain graphs encoding the representations of Sn. The 0-1 Conjecture states that the μ(x, w) ∈ {0, 1}. We present two counterexamples to this conjecture, the first in S16, for which x and w are in the same left cell, and the second in S10. The proof of the counterexample in S16 relies on computer calculations.

A Photographic Assignment for Abstract Algebra

AbstractWe study the polynomial

Quasisymmetric and Schur expansions of cycle index polynomials

\Phi \left( {x,y} \right) = \prod {_{j = 0}^{p - 1} \left( {1 - xw^j - yw^{qj} } \right)}