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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 deene a statistic on paths yielding two novel classes of polynomials. One… (More)

We define a bijection from Littlewood–Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials KλR(q) labeled by a partition λ and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of… (More)

- ANNE SCHILLING
- 2008

Hatayama et al. conjectured fermionic formulas associated with tensor products of U ′ q (g)-crystals B r,s. The crystals B r,s correspond to the Kirillov–Reshetikhin modules which are certain finite dimensional U ′ q (g)-modules. In this paper we present a combinatorial description of the affine crystals B r,1 of type D (1) n. A statistic preserving… (More)

- MASATO OKADO, ANNE SCHILLING
- 2003

Kerov, Kirillov, and Reshetikhin defined a bijection between highest weight vectors in the crystal graph of a tensor power of the vector representation , and combinatorial objects called rigged configurations, for type A (1) n. We define an analogous bijection for all nonexceptional affine types, thereby proving (in this special case) the fermionic formulas… (More)

- ANNE SCHILLING
- 2008

We give a review of the current status of the X = M conjecture. Here X stands for the one-dimensional configuration sum and M for the corresponding fermionic formula. There are three main versions of this conjecture: the unrestricted, the classically restricted and the level-restricted version. We discuss all three versions and illustrate the methods of… (More)

- Jason Bandlow, Anne Schilling, Mike Zabrocki
- J. Comb. Theory, Ser. A
- 2011

We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the… (More)

- MASATO OKADO, ANNE SCHILLING
- 2003

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… (More)

- Alexander Berkovich, Barry M. McCoy, Anne Schilling
- 1996

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 Bethe's Ansatz equations of the XXZ model and gives a general method to construct polynomial… (More)

- ANNE SCHILLING
- 2001

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 bijec-tion between Littlewood–Richardson tableaux and rigged configurations is well-behaved with respect to level-restriction and give an explicit characterization of… (More)

- Lipika Deka, Anne Schilling
- J. Comb. Theory, Ser. A
- 2006

A new fermionic formula for the unrestricted Kostka polynomials of type A (1) n−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–Reshetihkin modules, not just for the symmetric and anti-symmetric case. The fermionic formula can be interpreted in terms of a new set of… (More)