## Infinite Analysis Seminar Tokyo

Seminar information archive ～11/05｜Next seminar｜Future seminars 11/06～

Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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### 2009/01/24

11:00-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

一般化されたヤング図形の q-Hook formula

Catalan numbers and level 2 weight structures of $A^{(1)}_{p-1}$

On a dimer model with impurities

**仲田 研登**(京大数研) 11:00-12:00一般化されたヤング図形の q-Hook formula

[ Abstract ]

Young図形における hook formula は、組合せ論的には、その Young 図形の standar

d tableau の総数を数え上げる公式である。R. P. Stanley は reverse plane parti

tion のなす母関数を考えることにより、この公式をq-hook formula に拡張し、E. R

. Gansner はそれをさらに多変数に一般化した。

本講演では、この(多変数)q-Hook formula が(D. Peterson、R. A. Proctor の意

味の)一般化されたYoung図形においても成り立つこと紹介する。特にこれはPeterso

n の hook formula の証明も与える。

Young図形における hook formula は、組合せ論的には、その Young 図形の standar

d tableau の総数を数え上げる公式である。R. P. Stanley は reverse plane parti

tion のなす母関数を考えることにより、この公式をq-hook formula に拡張し、E. R

. Gansner はそれをさらに多変数に一般化した。

本講演では、この(多変数)q-Hook formula が(D. Peterson、R. A. Proctor の意

味の)一般化されたYoung図形においても成り立つこと紹介する。特にこれはPeterso

n の hook formula の証明も与える。

**土岡 俊介**(京大数研) 13:30-14:30Catalan numbers and level 2 weight structures of $A^{(1)}_{p-1}$

[ Abstract ]

Motivated by a connection between representation theory of

the degenerate affine Hecke algebra of type A and

Lie theory associated with $A^{(1)}_{p-1}$, we determine the complete

set of representatives of the orbits for the Weyl group action on

the set of weights of level 2 integrable highest weight representations of $\\widehat{\\mathfrak{sl}}_p$.

Applying a crystal technique, we show that Catalan numbers appear in their weight multiplicities.

Here "a crystal technique" means a result based on a joint work with S.Ariki and V.Kreiman,

which (as an application of the Littelmann's path model) combinatorially characterize

the connected component (usually called Kleshchev bipartition in the representation theoretic context)

$B(\\Lambda_0+\\Lambda_s)\\subseteq B(\\Lambda_0)\\otimes B(\\Lambda_s)$ in the tensor product.

Motivated by a connection between representation theory of

the degenerate affine Hecke algebra of type A and

Lie theory associated with $A^{(1)}_{p-1}$, we determine the complete

set of representatives of the orbits for the Weyl group action on

the set of weights of level 2 integrable highest weight representations of $\\widehat{\\mathfrak{sl}}_p$.

Applying a crystal technique, we show that Catalan numbers appear in their weight multiplicities.

Here "a crystal technique" means a result based on a joint work with S.Ariki and V.Kreiman,

which (as an application of the Littelmann's path model) combinatorially characterize

the connected component (usually called Kleshchev bipartition in the representation theoretic context)

$B(\\Lambda_0+\\Lambda_s)\\subseteq B(\\Lambda_0)\\otimes B(\\Lambda_s)$ in the tensor product.

**中野 史彦**(高知大理学部数学) 15:00-16:00On a dimer model with impurities

[ Abstract ]

We consider the dimer problem on a non-bipartite graph $G$, where there are two types of dimers one of which we regard impurities. Results of simulations using Markov chain seem to indicate that impurities are tend to distribute on the boundary, which we set as a conjecture. We first show that there is a bijection between the set of dimer coverings on

$G$ and the set of spanning forests on two graphs which are made from $G$, with configuration of impurities satisfying a pairing condition, and this bijection can be regarded as a extension of the Temperley bijection. We consider local move consisting of two operations, and by using the bijection mentioned above, we prove local move connectedness. Finally, we prove that the above conjecture is true,

in some spacial cases.

We consider the dimer problem on a non-bipartite graph $G$, where there are two types of dimers one of which we regard impurities. Results of simulations using Markov chain seem to indicate that impurities are tend to distribute on the boundary, which we set as a conjecture. We first show that there is a bijection between the set of dimer coverings on

$G$ and the set of spanning forests on two graphs which are made from $G$, with configuration of impurities satisfying a pairing condition, and this bijection can be regarded as a extension of the Temperley bijection. We consider local move consisting of two operations, and by using the bijection mentioned above, we prove local move connectedness. Finally, we prove that the above conjecture is true,

in some spacial cases.