Related papers: Schur Polynomials do not have small formulas if th…

we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable…

The central open question of algebraic complexity is whether VP is unequal to VNP, which is saying that the permanent cannot be represented by families of polynomial-size algebraic circuits. For symmetric algebraic circuits, this has been…

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified…

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $\pi$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},\pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the…

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1,x2,...]. We suggest the "prism tableau model" for these polynomials. A novel aspect of this alternative to earlier results is that it directly…

The fact that Schubert polynomials are the weighted counting functions for reduced RC-graphs, also known as reduced pipe dreams, was established using their generating functions inside an appropriate Demazure algebra. Here we investigate…

We examine and present new combinatorics for the Schur polynomials from the viewpoint of quantum integrability. We introduce and analyze an integrable six-vertex model which can be viewed as a certain degeneration model from a t-deformed…

Let X be the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of X. We use these polynomials to…

We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron…

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several…

In light of recently proposed quantum algorithms that incorporate symmetries in the hope of quantum advantage, we show that with symmetries that are restrictive enough, classical algorithms can efficiently emulate their quantum counterparts…

We give a pattern-avoidance characterization of $w \in S_n$ such that the Schubert polynomial $\mathfrak{S}_w$ is a standard elementary monomial. This characterization tells us which quantum Schubert polynomials are easiest to compute. We…

We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite…

The Jack symmetric polynomials $P_\lambda^{(\alpha)}$ form a class of symmetric polynomials which are indexed by a partition $\lambda$ and depend rationally on a parameter $\alpha$. They reduced to the Schur polynomials when $\alpha=1$, and…

We prove that every supersymmetric Schur polynomial has a saturated Newton polytope (SNP). Our approach begins with a tableau-theoretic description of the support, which we encode as a polyhedron with a totally unimodular constraint matrix.…

Let $S(m|n,d)$ be the Schur superalgebra whose supermodules correspond to the polynomial representations of the supergroup $GL(m|n)$ of degree $d$. In this paper we determine the representation type of these algebras (i.e. classify the ones…

We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the…

The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. This…

We obtain a new formula to relate the value of a Schur polynomial with variables $(x_1,\ldots,x_N)$ with values of Schur polynomials at $(1,\ldots,1)$. This allows to study the limit shape of perfect matchings on a square hexagon lattice…

For a positive integer $n$ and a real number $\alpha$, the generalized Laguerre polynomials are defined by \begin{align*} L^{(\alpha)}_n(x)=\sum^n_{j=0}\frac{(n+\alpha)(n-1+\alpha)\cdots (j+1+\alpha)(-x)^j}{j!(n-j)!}. \end{align*} These…