## Infinite Analysis Seminar Tokyo

Seminar information archive ～10/11｜Next seminar｜Future seminars 10/12～

Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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### 2014/11/20

15:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

An explicit formula for the specialization of nonsymmetric

Macdonald polynomials at $t = \infty$ (JAPANESE)

divisor function and strict partition (JAPANESE)

**Fumihiko Nomoto**(Department of Mathematics, Tokyo Institute of Technology, Graduate school of science and Engineering) 15:00-16:30An explicit formula for the specialization of nonsymmetric

Macdonald polynomials at $t = \infty$ (JAPANESE)

[ Abstract ]

Orr-Shimozono obtained an explicit formula for nonsymmetric Macdonald polynomials with Hecke parameter $t$ set to $\infty$, which is described in terms of an affine root system

and an affine Weyl group. On the basis of this work, we give another explicit formula for the specialization above, which is described in terms of the quantum Bruhat graph associated with a finite root system and a finite Weyl group.

More precisely, we interpret the specialization above as the graded character of an explicitly specified set of quantum Lakshmibai-Seshadri (LS) paths. Here we note that the set of quantum LS paths (of a given shape) provides an explicit realization of the crystal basis of a quantum Weyl module over the quantum affine algebra.

In this talk, I will explain our explicit formula

by exhibiting a few examples.

Also, I will give an outline of the proof.

Orr-Shimozono obtained an explicit formula for nonsymmetric Macdonald polynomials with Hecke parameter $t$ set to $\infty$, which is described in terms of an affine root system

and an affine Weyl group. On the basis of this work, we give another explicit formula for the specialization above, which is described in terms of the quantum Bruhat graph associated with a finite root system and a finite Weyl group.

More precisely, we interpret the specialization above as the graded character of an explicitly specified set of quantum Lakshmibai-Seshadri (LS) paths. Here we note that the set of quantum LS paths (of a given shape) provides an explicit realization of the crystal basis of a quantum Weyl module over the quantum affine algebra.

In this talk, I will explain our explicit formula

by exhibiting a few examples.

Also, I will give an outline of the proof.

**Masanori Ando**(Wakkanai Hokusei Gakuen University) 17:00-18:30divisor function and strict partition (JAPANESE)

[ Abstract ]

We know that the q-series identity of Uchimura-type is related with the divisor function.

It is obtained also as a specialization of basic hypergeometric series.

In this seminar, we interprete this identity from the point of view of combinatorics of partitions of integers.

We give its proof by using the mock involution map.

We know that the q-series identity of Uchimura-type is related with the divisor function.

It is obtained also as a specialization of basic hypergeometric series.

In this seminar, we interprete this identity from the point of view of combinatorics of partitions of integers.

We give its proof by using the mock involution map.