Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara’s theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials. One of these provides a generalization o...

The Kostka-Foulkes polynomials K λ,μ(q) related to a root system φ can be defined as alternated sums running over the Weyl group associated to φ. By restricting these sums over the elements of the symmetric group when φ is of type Bn, Cn orDn, we obtain again a class K̃ φ λ,μ(q) of Kostka-Foulkes polynomials. When φ is of type Cn or Dn there exists a duality beetween these polynomials and some n...

The hive model is used to explore the properties of both Kostka coefficients and stretched Kostka coefficient polynomials. It is shown that both of these may factorise, and that they can then be expressed as products of certain primitive coefficients and polynomials, respectively. It is further shown how to determine a sequence of linear factors (t+m) of the primitive polynomials, where t is th...

We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov’s log-concavity conjecture for Littlewood–Richardson coefficients in the special case of Kostka numbers.

The Kostka-Foulkes polynomials K λ,μ(q) related to a root system φ can be defined as alternated sums running over the Weyl group associated to φ. By restricting these sums over the elements of the symmetric group when φ is of type Bn, Cn orDn, we obtain again a class K̃ φ λ,μ(q) of Kostka-Foulkes polynomials. When φ is of type Cn or Dn there exists a duality beetween these polynomials and some n...

For any triple $(i,a,\mu)$ consisting of a vertex $i$ in quiver $Q$, positive integer $a$, and dominant $GL_a$-weight $\mu$, we define current $H^{(i,a)}_\mu$ acting on the tensor power $\Lambda^Q$ symmetric functions over vertices $Q$. These provide generalization parabolic Garsia-Jing creation operators theory Hall-Littlewood functions. $(\mathbf{i},\mathbf{a},\mu(\bullet))$ sequences such da...

We will look at the Catalan numbers from the Rigged Configurations point of view originated [9] from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models . Our strategy is to take a combinatorial interpretation of Catalan numbers Cn as the number of standard Young tableaux of rectangular shape (n2), or equivalently, as the Kostka ...

Using tools from combinatorics, convex geometry and symplectic geometry, we study the behavior of the Kostka numbers Kλβ and Littlewood-Richardson coefficients cλμ (the type A weight multiplicities and Clebsch-Gordan coefficients). We show that both are given by piecewise polynomial functions in the entries of the partitions and compositions parametrizing them, and that the domains of polynomia...

A combinatorial proof of the unimodality of the generalized q-Gaussian coefficients [ N λ ] q based on the explicit formula for Kostka-Foulkes polynomials is given. 1. Let us mention that the proof of the unimodality of the generalized Gaussian coefficients based on theoretic-representation considerations was given by E.B. Dynkin [1] (see also [2], [10], [11]). Recently K.O’Hara [6] gave a cons...

Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Eğecioğlu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, ...