Anne Schilling

Inhomogeneous Lattice Paths, Generalized Kostka Polynomials and A n −1 Supernomials

Abstract: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 Kashiwaras theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials. One of these provides a generalization of the Kostka polynomials, while the other, which we name the An−1 supernomial, is a q-deformation of the expansion coefficients of products of Schur polynomials. Many well-known results for Kostka polynomials are extended leading to representations of our polynomials in terms of a charge statistic on Littlewood–Richardson tableaux and in terms of fermionic configuration sums. Several identities for the generalized Kostka polynomials and the An−1 supernomials are proven or conjectured. Finally, a connection between the supernomials and Baileys lemma is made.

Rogers--Schur--Ramanujan Type Identities for the M(p,p') minimal models of Conformal Field Theory

Abstract:We present and prove Rogers–Schur–Ramanujan (Bose/Fermi) type identities for the Virasoro characters of the minimal model M(p,p′). The proof uses the continued fraction decomposition of p′/p introduced by Takahashi and Suzuki for the study of the Bethes Ansatz equations of the XXZ model and gives a general method to construct polynomial generalizations of the fermionic form of the characters which satisfy the same recursion relations as the bosonic polynomials of Forrester and Baxter. We use this method to get fermionic representations of the characters for many classes of r and s.

Existence of Kirillov–Reshetikhin crystals for nonexceptional types

Author(s): Okado, Masato; Schilling, Anne | Abstract: Using the methods of Kang et al. and recent results on the characters of Kirillov-Reshetikhin modules by Nakajima and Hernandez, the existence of Kirillov-Reshetikhin crystals B^{r,s} is established for all nonexceptional affine types. We also prove that the crystals B^{r,s} of type B_n^{(1)}, D_n^{(1)}, and A_{2n-1}^{(2)} are isomorphic to recently constructed combinatorial crystals for r not a spin node.

Kirillov-Reshetikhin crystals for nonexceptional types

We provide combinatorial models for all Kirillov--Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types D_n^(1), B_n^(1), A_{2n-1}^(2) we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type C_n^(1) we use a Dynkin diagram folding and for types A_{2n}^(2), D_{n+1}^(2) a similarity construction. We also show that for types C_n^(1) and D_{n+1}^(2) the analog of the Dynkin diagram automorphism exists on the level of crystals.

Fermionic Formulas for Level-Restricted Generalized Kostka Polynomials and Coset Branching Functions

Abstract: Level-restricted paths play an important rôle in crystal theory. They correspond to certain highest weight vectors of modules of quantum affine algebras. We show that the recently established bijection between Littlewood–Richardson tableaux and rigged configurations is well-behaved with respect to level-restriction and give an explicit characterization of level-restricted rigged configurations. As a consequence a new general fermionic formula for the level-restricted generalized Kostka polynomial is obtained. Some coset branching functions of type A are computed by taking limits of these fermionic formulas.

X = M FOR SYMMETRIC POWERS

Author(s): Schilling, Anne; Shimozono, Mark | Abstract: The X=M conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum X expressed as the generating function of crystal paths with energy statistics and the fermionic formula M for all affine Kac--Moody algebra. In this paper we prove the X=M conjecture for tensor products of Kirillov--Reshetikhin crystals B^{1,s} associated to symmetric powers for all nonexceptional affine algebras.

New fermionic formula for unrestricted Kostka polynomials

A new fermionic formula for the unrestricted Kostka polynomials of type An-1(1) is presented. This formula is different from the one given by Hatayama et al. and is valid for all crystal paths based on Kirillov-Reshetikhin modules, not just for the symmetric and antisymmetric case. The fermionic formula can be interpreted in terms of a new set of unrestricted rigged configurations. For the proof a statistics preserving bijection from this new set of unrestricted rigged configurations to the set of unrestricted crystal paths is given which generalizes a bijection of Kirillov and Reshetikhin.

A Uniform Model for Kirillov–Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph

© 2014

Virtual crystals and Kleber's algorithm

Abstract: Kirillov and Reshetikhin conjectured what is now known as the fermionic formula for the decomposition of tensor products of certain finite dimensional modules over quantum affine algebras. This formula can also be extended to the case of q-deformations of tensor product multiplicities as recently conjectured by Hatayama et al. In its original formulation it is difficult to compute the fermionic formula efficiently. Kleber found an algorithm for the simply-laced algebras which overcomes this problem. We present a method which reduces all other cases to the simply-laced case using embeddings of affine algebras. This is the fermionic analogue of the virtual crystal construction by the authors, which is the realization of crystal graphs for arbitrary quantum affine algebras in terms of those of simply-laced type.

Virtual crystals and fermionic formulas of type _{+1}⁽²⁾, _{2}⁽²⁾, and _{}⁽¹⁾

A device having utility in interfering with a persons desire to hold an object such as a cigarette, cigar or pipe, between his lips, i.e., an anti-smoking device. In the form of a cigarette holder, it includes a generally tubular shell having first and second ends, with the first end being adapted to receive a cigarette; the second end thereof includes structure adapted to be held between a persons lips. A DC voltage source (such as a dry cell battery) of at least six volts and preferably nine volts is mounted within the shell. First and second electrically conductive members are connected to the output of the DC source, with the distal ends of said conductive members extending alongside the lip-contacting structure so that they may be readily touched by a persons lips. The distal ends of said conductive members are separated so as to form a normally open electrical path, such that placing the lip-contacting structure between a persons lips will instantaneously close the electrical path and result in the discharge of DC current from said source through the lips. A potentiometer is optionally provided to adjust the flow of current from a minimum of about one milliamp (in order to be discernable) to a maximum of about five milliamps (so as to avoid intolerable sensations). Additionally, means are disclosed for re-charging a battery which is permanently mounted within the shell of a cigarette holder or the like.