q-Supernomial coefficients: From riggings to ribbons

q-Supernomial coefficients are generalizations of the q-binomial coefficients. They can be defined as the coefficients of the Hall-Littlewood symmetric function in a product of the complete symmetric functions or the elementary symmetric functions. Hatayama et al. give explicit expressions for these q-supernomial coefficients. A combinatorial expression as the generating function of ribbon tableaux with (co)spin statistic follows from the work of Lascoux, Leclerc and Thibon. In this paper we interpret the formulas by Hatayama et al. in terms of rigged configurations and provide an explicit statistic preserving bijection between rigged configurations and ribbon tableaux thereby establishing a new direct link between these combinatorial objects.Comment: 19 pages, svcon2e.sty file require

Rigged configurations and the Bethe Ansatz

These notes arose from three lectures presented at the Summer School on Theoretical Physics "Symmetry and Structural Properties of Condensed Matter" held in Myczkowce, Poland, on September 11-18, 2002. We review rigged configurations and the Bethe Ansatz. In the first part, we focus on the algebraic Bethe Ansatz for the spin 1/2 XXX model and explain how rigged configurations label the solutions of the Bethe equations. This yields the bijection between rigged configurations and crystal paths/Young tableaux of Kerov, Kirillov and Reshetikhin. In the second part, we discuss a generalization of this bijection for the symmetry algebra $D_n^{(1)}$, based on work in collaboration with Okado and Shimozono.Comment: 24 pages; lecture notes; axodraw style file require

A bijection between type D_n^{(1)} crystals and rigged configurations

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_n^{(1)}. A statistic preserving bijection between crystal paths for these crystals and rigged configurations is given, thereby proving the fermionic formula in this case. This bijection reflects two different methods to solve lattice models in statistical mechanics: the corner-transfer-matrix method and the Bethe Ansatz.Comment: 38 pages; version to appear in J. Algebr

Crystal structure on rigged configurations

Rigged configurations are combinatorial objects originating from the Bethe Ansatz, that label highest weight crystal elements. In this paper a new unrestricted set of rigged configurations is introduced for types ADE by constructing a crystal structure on the set of rigged configurations. In type A an explicit characterization of unrestricted rigged configurations is provided which leads to a new fermionic formula for unrestricted Kostka polynomials or q-supernomial coefficients. The affine crystal structure for type A is obtained as well.Comment: 20 pages, 1 figure, axodraw and youngtab style file necessar

Richard Stanley through a crystal lens and from a random angle

We review Stanley's seminal work on the number of reduced words of the longest element of the symmetric group and his Stanley symmetric functions. We shed new light on this by giving a crystal theoretic interpretation in terms of decreasing factorizations of permutations. Whereas crystal operators on tableaux are coplactic operators, the crystal operators on decreasing factorization intertwine with the Edelman-Greene insertion. We also view this from a random perspective and study a Markov chain on reduced words of the longest element in a finite Coxeter group, in particular the symmetric group, and mention a generalization to a poset setting.Comment: 11 pages; 3 figures; v2 updated references and added discussion on Coxeter-Knuth grap

Finite-Dimensional Crystals B^{2,s} for Quantum Affine Algebras of type D_{n}^{(1)}

The Kirillov--Reshetikhin modules W^{r,s} are finite-dimensional representations of quantum affine algebras U'_q(g), labeled by a Dynkin node r of the affine Kac--Moody algebra g and a positive integer s. In this paper we study the combinatorial structure of the crystal basis B^{2,s} corresponding to W^{2,s} for the algebra of type D_n^{(1)}.Comment: 34 pages; final version to appear in J. Alg. Combi

Promotion operator on rigged configurations of type A

Recently, the analogue of the promotion operator on crystals of type A under a generalization of the bijection of Kerov, Kirillov and Reshetikhin between crystals (or Littlewood--Richardson tableaux) and rigged configurations was proposed. In this paper, we give a proof of this conjecture. This shows in particular that the bijection between tensor products of type A_n^{(1)} crystals and (unrestricted) rigged configurations is an affine crystal isomorphism.Comment: 37 page

A Demazure crystal construction for Schubert polynomials

Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by Morse and the second author in connection with Stanley symmetric functions.Comment: 18 pages, 16 figures; version 2: references added and update

Crystal structure on rigged configurations and the filling map

In this paper, we extend work of the first author on a crystal structure on rigged configurations of simply-laced type to all non-exceptional affine types using the technology of virtual rigged configurations and crystals. Under the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin crystals specialized to a single tensor factor, we obtain a new tableaux model for Kirillov-Reshetikhin crystals. This is related to the model in terms of Kashiwara-Nakashima tableaux via a filling map, generalizing the recently discovered filling map in type $D_n^{(1)}$.Comment: 45 page

Unified theory for finite Markov chains

We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup $S$. Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to $P(n)$, the set of all subsets of an $n$ element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or $\mathscr{R}$-trivial bands) by Brown and Diaconis, extensions to $\mathscr{R}$-trivial semigroups by Ayyer, Steinberg, Thi\'ery and the second author, and important recent work by Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of $S$ in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded $S$. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded $S$ and the original $S$. Analyzing the normal forms also provides an estimate on the mixing time.Comment: 29 pages, 12 figures; v2: Section 3.2 added, references added, revision of introduction, title change; v3: typos fixed and clarifications adde