An extended Schur's lemma and its application

Schur Function Expansions and the Rational Shuffle Theorem

Gorsky and Negut introduced operators $Q_{m,n}$ on symmetric functions and conjectured that, in the case where $m$ and $n$ are relatively prime, the expression ${Q}_{m,n}(1)$ is given by the Hikita polynomial ${H}_{m,n}[X;q,t]$. Later, Bergeron-Garsia-Leven-Xin extended and refined the conjectures of ${Q}_{m,n}(1)$ for arbitrary $m$ and $n$ which we call the Extended Rational Shuffle Conjecture. In the special case ${Q}_{n+1,n}(1)$, the Rational Shuffle Conjecture becomes the Shuffle Conjecture of Haglund-Haiman-Loehr-Remmel-Ulyanov, which was proved in 2015 by Carlsson and Mellit as the Shuff…

Codes and shifted codes

The action of the Bernstein operators on Schur functions was given in terms of codes in [CG] and extended to the analog in Schur Q-functions in [HJS]. We define a new combinatorial model of extended codes and show that both of these results follow from a natural combinatorial relation induced on codes. The new algebraic structure provides a natural setting for Schur functions indexed by compositions.

Extended commonality of paths and cycles via Schur convexity

A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring, or equivalently, $t_H(W)+t_H(1-W)\geq 2^{1-e(H)}$ holds for every graphon $W:[0,1]^2\rightarrow [0,1]$, where $t_H(.)$ denotes the homomorphism density of the graph $H$. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989). We prove a graph homomorphism inequality that extends the commonality of paths and cycl…

The Horn recursion for Schur P- and Q- functions: Extended Abstract

A consequence of work of Klyachko and of Knutson-Tao is the Horn recursion to determine when a Littlewood-Richardson coefficient is non-zero. Briefly, a Littlewood-Richardson coefficient is non-zero if and only if it satisfies a collection of Horn inequalities which are indexed by smaller non-zero Littlewood-Richardson coefficients. There are similar Littlewood-Richardson numbers for Schur P- and Q- functions. Using a mixture of combinatorics of root systems, combinatorial linear algebra in Lie algebras, and the geometry of certain cominuscule flag varieties, we give Horn recursions to determi…

旗空间、拟阵舒尔代数与斯坦伯格表示

Flag Space, Matroidal Schur Algebras and the Steinberg Representation

研究通过生成元和关系定义拟阵舒尔代数的扩展版本，在射影几何中关联其与\(GL(n,q)\)模斯坦伯格表示结构，一般情形下证明其有三角或里迪分解，还得到原始代数新描述并阐明与旗空间联系。

Extended Schur's $Q$-functions and the full Kostant--Toda hierarchy on the Lie algebra of type $D$

The full Kostant--Toda hierarchy on a semisimple Lie algebra is a system of Lax equations, in which the flows are determined by the gradients of the Chevalley invariants.This paper is concerned with the full Kostant--Toda hierarchy on the even orthogonal Lie algebra. By using a Pfaffian of the Lax matrix as one of the Chevalley invariants, we construct an explicit form of the flow associated to this invariant. As a main result, we introduce an extension of the Schur's $Q$-functions in the time variables, and use them to give explicit formulas for the polynomial $τ$-functions of the hierarchy.

The immaculate basis of the non-commutative symmetric functions

FPSAC 2013 Extended Abstract. We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula.

Bernstein operators and super-Schur functions: combinatorial aspects

The Bernstein vertex operators, which can be used to build recursively the Schur functions, are extended to superspace. Four families of super vertex operators are defined, corresponding to the four natural families of Schur functions in superspace. Combinatorial proofs that the super Bernstein vertex operators indeed build the Schur functions in superspace recursively are provided. We briefly mention a possible realization, in terms of symmetric functions in superspace, of the super-KP hierarchy, where the tau-function naturally expands in one of the super-Schur bases.

Weak Bruhat interval modules of the 0-Hecke algebra

The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a $0$-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the $0$-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restri…