Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A

We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a corresponding cyclotomic rational Cherednik algebra. Besides, it appears to be closely related to the Harish-Chandra branching graph for the appropriate finite unitary group, according to [8]. In this paper, we make explicit a particular isomorphism between connected components of the crystal graphs of Fock spaces. This so-called "canonical" crystal isomorphism turns out to be expressible only in terms of: - Schensted's classic bumping procedure, - the cyclage isomorphism defined in [13], - a new crystal isomorphism, easy to describe, acting on cylindric multipartitions. We explain how this can be seen as an analogue of the bumping algorithm for affine type $A$. Moreover, it yields a combinatorial characterisation of the vertices of any connected component of the crystal of the Fock space

A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule

We study the A_k generalized model of the O(1) loop model on a cylinder. The affine Hecke algebra associated with the model is characterized by a vanishing condition, the cylindric relation. We present two representations of the algebra: the first one is the spin representation, and the other is in the vector space of states of the A_k generalized model. A state of the model is a natural generalization of a link pattern. We propose a new graphical way of dealing with the Yang-Baxter equation and $q$-symmetrizers by the use of the rhombus tiling. The relation between two representations and the meaning of the cylindric relations are clarified. The sum rule for this model is obtained by solving the q-KZ equation at the Razumov-Stroganov point.Comment: 43 pages, 22 figures, LaTeX, (ver 2) Introduction rewritten and Section 4.3 adde

Cylindric Reverse Plane Partitions and 2D TQFT

The ring of symmetric functions carries the structure of a Hopf algebra. When computing the coproduct of complete symmetric functions $h_\lambda$ one arrives at weighted sums over reverse plane partitions (RPP) involving binomial coefficients. Employing the action of the extended affine symmetric group at fixed level $n$ we generalise these weighted sums to cylindric RPP and define cylindric complete symmetric functions. The latter are shown to be $h$-positive, that is, their expansions coefficients in the basis of complete symmetric functions are non-negative integers. We state an explicit formula in terms of tensor multiplicities for irreducible representations of the generalised symmetric group. Moreover, we relate the cylindric complete symmetric functions to a 2D topological quantum field theory (TQFT) that is a generalisation of the celebrated $\mathfrak{\widehat{sl}}_n$-Verlinde algebra or Wess-Zumino-Witten fusion ring, which plays a prominent role in the context of vertex operator algebras and algebraic geometry.Comment: 13 pages, 1 figure, accepted conference proceedings article for FPSAC2018 (Hanover

Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group W is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied in Boothby et al.. In particular the authors enumerated cyclically fully commutative elements in all Coxeter groups having a finite number of them. In this work we characterize and enumerate cyclically fully commutative elements according to their Coxeter length in all finite or affine Coxeter groups by using a new operation on heaps, the cylindric transformation. In finite types, this refines the work of Boothby et al., by adding a new parameter. In affine type, all the results are new. In particular, we prove that there is a finite number of cyclically fully commutative logarithmic elements in all affine Coxeter groups. We study afterwards the cyclically fully commutative involutions and prove that their number is finite in all Coxeter groups.Comment: 23 pages, 16 figure