Related papers: Weak Bruhat interval modules for genomic Schur fun…

In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood-Richardson coefficients in torus-equivariant $K$-theory of Grassmannians. We then studied the genomic Schur…

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…

Genomic Schur functions were introduced by Pechenik and Yong in connection with the $K$-theory of Grassmannians. Pechenik proved that genomic Schur functions admit a positive expansion in the basis of fundamental quasisymmetric functions…

The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial…

We begin by deriving an action of the 0-Hecke algebra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristics are the natural refinements of Schur functions known as…

We construct modules of the $0$-Hecke algebra whose images under the quasisymmetric characteristic map are the Young row-strict quasisymmetric Schur functions. This provides a representation-theoretic interpretation of this basis of…

We explain how genomic tableaux [Pechenik-Yong '15] are a semistandard complement to increasing tableaux [Thomas-Yong '09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender-Knuth…

For a skew shape $\lambda/\mu$, we define the hybrid Grothendieck polynomial $${G}_{\lambda/\mu}(\textbf{x};\textbf{t};\textbf{w}) =\sum_{T\in \mathrm{SVRPP}(\lambda/\mu)} \textbf{x}^{\mathrm{ircont}(T)}\textbf{t}^{\mathrm{ceq}…

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials,…

We introduce and study a generalization $s_{(\mu|\lambda)}$ of the Schur functions called the almost symmetric Schur functions. These functions simultaneously generalize the finite variable key polynomials and the infinite variable Schur…

The rational Cherednik algebra $\HH$ is a certain algebra of differential-reflection operators attached to a complex reflection group $W$. Each irreducible representation $S^\lambda$ of $W$ corresponds to a standard module $M(\lambda)$ for…

Consider the Cauchy problem for a strictly hyperbolic, $N\times N$ quasilinear system in one space dimension $$ u_t+A(u) u_x=0,\qquad u(0,x)=\bar u(x), \eqno (1) $$ where $u \mapsto A(u)$ is a smooth matrix-valued map, and the initial data…

Let $\alpha$ range over the set of compositions. Dual immaculate quasisymmetric functions $\mathfrak{S}_\alpha^*$, introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki, provide a quasisymmetric analogue of Schur functions. They also…

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…

For an integer partition $ \lambda$ of $n$ and an $n \times n$ matrix $A$, consider the expansion of the immanant $\text{Imm}^{\lambda}(A)$ as a sum indexed by permutations $\sigma$ of order $n$, with coefficients given by the irreducible…

We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_\lambda$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive…

We consider the expansion of the square of a complete homogeneous function $h_\lambda$, or of an elementary symmetric function $e_\lambda$, in the basis of Schur functions. This square also decomposes into two plethysms, $s_2[h_\lambda]$…

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the…

We study the $H_n(0)$-module $\mathbf{S}^\sigma_\alpha$ due to Tewari and van Willigenburg, which was constructed using new combinatorial objects called standard permuted composition tableaux and decomposed into cyclic submodules. First, we…

In the present paper we investigate a new class of infinite-dimensional modules over the hyperalgebra of a semi-simple algebraic group in positive chararacteristic called quasi-Verma modules. We provide a purely algebraic construction of…