Related papers: Lower bounds for Kazhdan-Lusztig polynomials from …

We derive a tight upper bound on the probability over $\mathbf{x}=(x_1,\dots,x_\mu) \in \mathbb{Z}^\mu$ uniformly distributed in $ [0,m)^\mu$ that $f(\mathbf{x}) = 0 \bmod N$ for any $\mu$-linear polynomial $f \in…

In this article we make several contributions of independent interest. First, we introduce the notion of stressed hyperplane of a matroid, essentially a type of cyclic flat that permits to transition from a given matroid into another with…

We prove the combinatorial invariance of the coefficient of $q$ in Kazhdan--Lusztig polynomials for arbitrary Coxeter groups. As a result, we obtain the Combinatorial Invariance Conjecture, of Lusztig and of Dyer, also for Bruhat intervals…

For any large prime $q$, $x \leq 1$ and any real $k\geq 2$, we prove a lower bound for the following $2k$-th moment \begin{equation*} \sum_{\substack{\chi \in X_q^*}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k}, \end{equation*} where…

Let $V,W$ be representations of a cyclic group $G$ of prime order $p$ over a field $k$ of characteristic $p$. The module of covariants $k[V,W]^G$ is the set of $G$-equivariant polynomial maps $V \rightarrow W$, and is a module over…

For a polynomial $f(x)\in \mathbb Z[x]$ we study an analogue of Jacobsthal function, defined by the formula \[ j_f(N)=\max_{m}\{\text{For some } x\in \mathbb N \text{ the inequality } (x+f(i),N)>1 \text{ holds for all }i\leq m\}. \] We…

We explain a strategy for a proof of the positivity of all coefficients of Kazhdan-Lusztig-polynomials for arbitrary Coxeter groups by constructing spaces whose dimensions we conjecture to be these coefficients.

In this paper we compute the leading coefficients $\mu (u,w)$ of the Kazhdan--Lusztig polynomials $P_{u,w}$ for an affine Weyl group of type $\tilde{B}_2$. By using the \textbf{a}-function of a Coxeter group defined by Lusztig (see [L1,…

We consider the set $\Irr(W)$ of (complex) irreducible characters of a finite Coxeter group $W$. The Kazhdan--Lusztig theory of cells gives rise to a partition of $\Irr(W)$ into "families" and to a natural partial order $\leq_{\cLR}$ on…

We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a…

Let $W$ be a Coxeter group, and for $u,v\in W$, let $R_{u,v}(q)$ be the Kazhdan-Lusztig $R$-polynomial indexed by $u$ and $v$. In this paper, we present a combinatorial proof of the inversion formula on $R$-polynomials due to Kazhdan and…

This paper is motivated by several combinatorial actions of the affine Weyl group of type $C_n$. Addressing a question of David Vogan, it was shown in an earlier paper that these permutation representations are essentialy…

We develop a marking system for an analog of Hasse diagrams of intervals $[u,v]$ with $u\leq v$ in a Hermitian symmetric pair $W/W_J$, and use this to create a closed form algorithm for computing relative R-polynomials. The uniform nature…

We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple…

Lusztig proved that the Kazhdan-Lusztig basis of a spherical Hecke algebra can be essentially identified with the Weyl characters of the Langlands dual group. We generalize this result to the unequal parameter case. The new proof is pretty…

The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points of $S$ whose convex hull contains the origin in the interior.…

Let $\mathfrak{g}$ be a simple finite dimensional complex Lie algebra and let $\widehat{\mathfrak{g}}$ be the corresponding affine Lie algebra. Kac and Wakimoto observed that in some cases the coefficients in the character formula for a…

Kazhdan constants of discrete groups are hard to compute and the actual constants are known only for several classes of groups. By solving a semidefinite programming problem by a computer, we obtain a lower bound of the Kazhdan constant of…

We study the problem of list-decodable linear regression, where an adversary can corrupt a majority of the examples. Specifically, we are given a set $T$ of labeled examples $(x, y) \in \mathbb{R}^d \times \mathbb{R}$ and a parameter $0<…

We obtain explicit branching rules for graded cell modules and graded simple modules over the endomorphism algebra of a Bott-Samelson bimodule. These rules allow us to categorify a well-known recursive formula for Kazhdan-Lusztig…