Friday 30 September

We describe a vertex model for the Lascoux-Leclerc-Thibon (LLT) polynomials, whose weights are obtained from fusing the fundamental representations of the quantum affine superalgebra U_q (\hat{sl} (1|n)). We further explain how the structure of the latter can be used to both rederive old and prove new properties of LLT polynomials. This is joint work with Alexei Borodin and Michael Wheeler.

The box-ball system (BBS) is a cellular automaton that exhibits the solitonic behavior, introduced by [Takahashi-Satsuma-90]. It is known that the dynamics of BBS can be linearized by several methods. In this talk, we give the explicit relationships between two different methods, the rigged configuration (RC) and the slot decomposition (SD). To do so, we introduce the notion of the carrier process with seat numbers and the corresponding seat number configuration (SNC). We show that SNC itself gives a new linearization method, and that by translating RC and SD in the language of SNC, the relationships between RC and SD become clear. This talk is based on the joint work with Matteo Mucciconi, Tomohiro Sasamoto and Makiko Sasada.

RSK is a birational map.

I present a family of new solutions to the tetrahedron equation RLLL = LLLR, where L may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the q-oscillator or q-Weyl algebras. When the three L’s are associated with the q-oscillator algebra, R coincides with the known intertwiner of the quantized coordinate ring Aq(sl3). On the other hand, L’s based on the q-Weyl algebra lead to new R’s whose elements are either factorized or expressed as a terminating q-hypergeometric type series. The RRRR=RRRR type tetrahedron equation has also been confirmed in many cases.

(Based on a joint work with S. Matsuike and A. Yoneyama.)