Related papers: The Stanley Conjecture Revisited
We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of…
We provide direct proofs of product and coproduct formulae for Schur functions where the coefficients (Littlewood--Richardson coefficients) are defined as counting puzzles. The product formula includes a second alphabet for the Schur…
We consider Jack polynomials $J_\lambda$ and their shifted analogue $J^#_\lambda$. In 1989, Stanley conjectured that $\langle J_\mu J_\nu, J_\lambda \rangle$ is a polynomial with nonnegative coefficients in the parameter $\alpha$. In this…
We find and prove a factorization formula for certain Macdonald Littlewood-Richardson coefficients $c_{\lambda\mu}^{\nu}(q,t)$. Namely, we consider the case that the Kostka number $K_{\mu, \nu -\lambda}$ is $1$. This settles a particular…
This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack…
The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…
We introduce a super version of the Littlewood--Richardson rule for super Schur functions over signed alphabets. We give in particular combinatorial interpretations of the super Littlewood--Richardson coefficients using the properties of…
We give a probabilistic interpretation of the coefficients of the elementary symmetric function expansion of the chromatic quasisymmetric function for any unit interval graph. As a corollary, we prove the Stanley--Stembridge conjecture.
In this paper, we study shifted Schur functions $S_\mu^\star$, as well as a new family of shifted symmetric functions $\mathfrak{K}_\mu$ linked to Kostka numbers. We prove that both are polynomials in multi-rectangular coordinates, with…
We use dual equivalence to give a short, combinatorial proof that Stanley symmetric functions are Schur positive. We introduce weak dual equivalence, and use it to give a short, combinatorial proof that Schubert polynomials are key…
The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K.…
We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial…
We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-Theory of a group ring RG. It allows to treat twisted group…
We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mf{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila…
A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's…
We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kra\'skiewicz and developed by Lam and Serrano. We also introduce…
The stretched Littlewood-Richardson coefficient $c^{t\nu}_{t\lambda,t\mu}$ was conjectured by King, Tollu, and Toumazet to be a polynomial function in $t.$ It was shown to be true by Derksen and Weyman using semi-invariants of quivers.…
We continue the work begun by Mickler-Moll investigating the properties of the polynomial eigenfunctions of the Nazarov-Sklyanin quantum Lax operator. By considering products of these eigenfunctions, we produce a novel generalization of a…
The reduced Schur functions are studied. Their relations to the basic representation of $A^(1)_{r-1}$ and modular representations of the symmetric groups are clarified. Littlewood-Richardson coefficients appear in the linear relations among…
We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka--Foulkes polynomials) share symmetries related to the operations…