On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions

We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases.Comment: 42 pages, 2 figure

Yang-Baxter basis of Hecke algebra and Casselman's problem (extended abstract)

International audienceWe generalize the definition of Yang-Baxter basis of type A Hecke algebra introduced by A.Lascoux, B.Leclerc and J.Y.Thibon (Letters in Math. Phys., 40 (1997), 75–90) to all the Lie types and prove their duality. As an application we give a solution to Casselman's problem on Iwahori fixed vectors of principal series representation of p-adic groups

Factorial Schur functions and the Yang-Baxter equation

Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of "shift" parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expressed as the partition function of a particular statistical-mechanical system (six vertex model). The proof is based on the Yang-Baxter equation. There is a deformation parameter $t$ which may be specialized in different ways. If $t=-1$, then we recover the expression of the factorial Schur function as a ratio of alternating polynomials. If $t=0$, we recover the description as a sum over tableaux. If $t=\infty$ we recover a description of Lascoux that was previously considered by McNamara. We also are able to prove using the Yang-Baxter equation the asymptotic symmetry of the factorial Schur functions in the shift parameters. Finally, we give a proof using our methods of the dual Cauchy identity for factorial Schur functions. Thus using our methods we are able to give thematic proofs of many of the properties of factorial Schur functions