A combinatorial proof of a Weyl type formula for hook Schur polynomials

In this paper, we give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate step, we establish a formula for the multiplication of a Schubert class by a quantum Schur polynomial indexed by a hook partition. This entails a detailed analysis of chains and intervals in the quantum Bruhat order. This analysis allows us to use results of Leung--Li and of Postnikov to re...

Kazhdan-Lusztig polynomials are important and mysterious objects in representation theory. Here we present a new formula for their computation for symmetric groups based on the Bruhat graph. Our approach suggests a solution to the combinatorial invariance conjecture for symmetric groups, a well-known conjecture formulated by Lusztig and Dyer in the 1980s.

In this paper, we present a direct bijective proof of the hook-length formula for standard immaculate tableaux, which arose in the study of non-commutative symmetric functions. Our proof is along the spirit of Novelli, Pak and Stoyanovskii's combinatorial proof of the hook-length formula for standard Young tableaux.

The generalized Kazhdan-Lusztig polynomials for the finite dimensional irreducible representations of the general linear superalgebra are computed explicitly. Using the result we establish a one to one correspondence between the set of composition factors of an arbitrary $r$-fold atypical $gl_{m|n}$-Kac-module and the set of composition factors of some $r$-fold atypical $gl_{r|r}$-Kac-module. The result of Kazhdan-Lusztig polynomials is also applied to prove a conjectural cha...

In the present paper, we study the notion of the Schur multiplier $\mathcal{M}(L)$ of an $n$-Lie superalgebra $L$, and prove that $\dim \mathcal{M}(L) \leq \sum_{i=0}^{n} {m\choose{i}} \mathcal{L}(n-i,k)$, where $\dim L=(m|k)$, $\mathcal{L}(0,k)=1$ and $\mathcal{L}(t,k) = \sum_{j=1}^{t}{{t-1}\choose{j-1}} {k\choose j}$, for $1\leq t\leq n$. Moreover, we obtain an upper bound for the dimension of $\mathcal{M}(L)$ in which $L$ is a nilpotent $n$-Lie superalgebra with one-dimens...

A survey written for the upcoming "Handbook of Enumerative Combinatorics".

In this paper we introduce doubly symmetric functions, arising from the equivalence of particular linear combinations of Schur functions and hook Schur functions. We study algebraic and combinatorial aspects of doubly symmetric functions, in particular as they form a subalgebra of the algebra of symmetric functions. This subalgebra is generated by the odd power sum symmetric functions. One consequence is that a Schur function itself is doubly symmetric if and only if it is th...

We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and k-Schur functions.

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. This model can be viewed as a discrete counterpart of the Littelmann path model, and has several advantages. Our construction is give...

We give a combinatorial construction for the canonical bases of the $\pm$-parts of the quantum enveloping superalgebra $\bfU(\mathfrak{gl}_{m|n})$ and discuss their relationship with the Kazhdan-Lusztig bases for the quantum Schur superalgebras $\bsS(m|n,r)$ introduced in \cite{DR}. We will also extend this relationship to the induced bases for simple polynomial representations of $\bfU(\mathfrak{gl}_{m|n})$.