The $\hat W$-orbit of $ρ$, Kostant's formula for powers of the Euler product and affine Weyl groups as permutations of Z

A study of Kostant-Kumar modules via Littelmann paths

We study, by means of Littelmann's theory of paths, Kostant-Kumar modules (KK modules for short), which by definition are certain submodules of the tensor product of two irreducible integrable highest weight representations of a symmetrizable Kac-Moody algebra. Our main result is an identification of a path model for any KK module as a subset of the well known path model for the tensor product consisting of concatenations of Lakshmibai-Seshadri paths. The technical results about extremal elements in Coxeter groups that we formulate and prove en route and the technique of their proofs should be…

关于仿射李超代数及其模的顶点超代数的积分形式

On integral forms for vertex superalgebras associated with affine Lie superalgebras and their modules

本文研究仿射顶点超代数及其模的积分形式，通过构造仿射李超代数通用包络超代数的积分形式，并利用Chevalley基推广Kostant-Garland积分形式理论，进而构造基于仿射李超代数的顶点（算子）超代数及其模的积分形式，并探讨积分形式包含共形向量以及反梯度模中的积分形式。

Fixed-point-free involutions and Schur P-positivity

The orbits of the symplectic group acting on the type A flag variety are indexed by the fixed-point-free involutions in a finite symmetric group. The cohomology classes of the closures of these orbits have polynomial representatives $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions $\hat{F}^{\tt{FPF}}_z$, which are stable limits of the polynomials $\hat{\mathfrak{S}}^{\tt{FPF}}_z$, are Schur $P$-positive. To do so, we construct an analogue of the Lascoux-Schützenberger tree, an algebraic recurrence that comp…

Affine transitions for involution Stanley symmetric functions

We study a family of symmetric functions $\hat F_z$ indexed by involutions $z$ in the affine symmetric group. These power series are analogues of Lam's affine Stanley symmetric functions and generalizations of the involution Stanley symmetric functions introduced by Hamaker, Pawlowski, and the first author. Our main result is to prove a transition formula for $\hat F_z$ which can be used to define an affine involution analogue of the Lascoux-Schützenberger tree. Our proof of this formula relies on Lam and Shimozono's transition formula for affine Stanley symmetric functions and some new techni…

Multiplicities of maximal weights of the $\hat{s\ell}(n) $-module $V(kΛ_0)$

Consider the affine Lie algebra $\hat{s\ell}(n)$ with null root $δ$, weight lattice $P$ and set of dominant weights $P^+$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\hat{s\ell}(n)$-module with level $k \geq 1$ highest weight $kΛ_0$. Let $wt(V)$ denote the set of weights of $V(kΛ_0)$. A weight $μ\in wt(V)$ is a maximal weight if $μ+ δ\not\in wt(V)$. Let $max^+(kΛ_0)= max(kΛ_0)\cap P^+$ denote the set of maximal dominant weights which is known to be a finite set. In 2014, the authors gave the complete description of the set $max^+(kΛ_0)$. In subsequent pap…

What is a true spectra of a finite Fourier transform

In this paper we deal with a finite abelian group $G$ and the abstract Fourier transform ${\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$. Then, we consider $\tilde{j}\circ {\mathcal F}:{\mathbb C}^G\to {\mathbb C}^\hat{G}$ where $\tilde j:{\mathbb C}^\hat{G}\to {\mathbb C}^G$ is defined by the composition with a bijection $j:G\to \hat{G}$. ($\tilde j$ is a pullback of $j$.) In particular, we show that $(\tilde{j}\circ {\mathcal F})^2$ is a permutation if and only if $j$ is a group isomorphism. Then, we study how the spectra of $\tilde{j}\circ{\mathcal F}$ depends on the isomorphism $j$.