### 2017年度RIMS共同研究（公開型）「表現論と組合せ論」プログラム

#### 10月10日 (火)

13:30--14:30

「対称群の既約表現をより小さな対称群に制限していったとき, いつ次数が $p$ と素な既約成分が現れるか？」この問はゲームと表現の間の研究において生まれたものである. 元々, ゲームと表現の間に何か関係があるであろうことは, 1960 年代頃に佐藤幹夫先生によって予想され, 川中宣明先生らによって様々な研究がなされてきた. 講演者は上の問に対して部分的な結果を与えることで, ゲームと表現の間のある一つの関係を得た. 本講演では, この研究で現れた表現論的な組合せ論の側面, 具体的には対称群の既約表現をより小さな対称群に制限していったときの$p$-core tower の様子を紹介する.
14:45--15:45

Quantum invariants of knots and the Andrews-Gordon identities for $A_2$

16:00--17:00

Generalized root systems and generalized quantum groups
Coxeter introduced Coxeter groups in 1934, and he classified the finite Coxeter groups in 1935. Those are classified into $A_n$, $B_n=C_n$, $D_n$, $F_4$, $E_6$, $E_7$,$E_8$, $G_2$, $I_n$, $H_3$ and $H_4$. Coxeter groups appear in many areas of Algebra and Geometry. One of the areas is the representation theory of Lie algebras. Perhaps from 1970's, many researchers have considered that they had to need a notion being had to be called Coxeter groupoids' and being had to be applied for study of the representation theory of Lie superalgebras. It turned out that in 2000's, Coxeter groupoids' are also necessary for study of the generalized quantum groups. With collaborators, Hiroyuki Yamane has achieved the following results. (0) Presentations of finite and affine type Lie superalgebras and their quantum superalgebras (1991-2003) (1) Definition of Coxeter groupoids with axioms, and Matsumoto-type theorem of them (2008) (2) Shapovalov determinants of the generalized quantum groups (2010) (3) Classification of the finite-dimensional irreducible representations of the generalized quantum groups (2015) (4) Harish-Chandra type theorem of the center of the generalized quantum groups (2017) (5) Bruhat order of the Coxeter groupoids (2017)

#### 10月11日 (水)

9:30--10:30

Polyhedral realizations of crystal bases and perfect bases with positivity properties
The main object in this talk is a certain rational convex polytope whose lattice points give a polyhedral realization of a Demazure crystal. The speaker and Naito proved that this is identical to the Newton-Okounkov body of a Schubert variety associated with a specific valuation, which is defined algebraically to be a highest term valuation. In this talk, we see that, on a perfect basis with some positivity properties, this valuation is identical to the geometrically natural valuation coming from a sequence of specific subvarieties of the Schubert variety. The existence of such a perfect basis follows from a categorification of the negative part of the quantized enveloping algebra. As a corollary, we prove that the associated Newton-Okounkov bodies coincide through an explicit affine transformation. This talk is based on a joint work with H. Oya.
10:45--11:45

BMR freeness for icosahedral families
We verify the Brou\'e-Malle-Rouquier freeness conjecture for cyclotomic Hecke algebras of type $G_{17}$, $G_{18}$ and $G_{19}$.
13:30--14:30

abstract
14:45--15:45

On the expansion coefficients of Tau-function of the KP and BKP hierarchies

16:00--17:00

Pieri rules for factorial and symplectic Schur $Q$-functions
The classical Pieri rule describes a decomposition of the product of two Schur functions, one of which corresponds to a one-row partition. In this talk we consider Ivanov's factorial $Q$-functions $Q_\lambda (x|a)$ and symplectic $Q$-functions $Q^C_\lambda (x)$. We prove that the coefficients of $Q_\lambda (x|a)$ in the expansion of the product $Q_\mu (x|a) Q_{(r)}(x|0)$ are polynomials in the factorial parameters $a$ with nonnegative integer coefficients. And we give a combinatorial description of the coefficients of $Q^C_\lambda (x)$ in the product $Q^C_\mu (x) Q^C_{(r)}(x)$.

#### 10月12日 (木)

9:30--10:30

ヘッセンバーグ多様体と超平面配置
ヘッセンバーグ多様体は旗多様体の部分多様体であり、そのトポロジーは他分野と関連していることが知られている。本講演では、ヘッセンバーグ多様体と超平面配置との関連について紹介する。特に、次の３つの環が環同型であることについて紹介する。(1) 正則な冪零ヘッセンバーグ多様体のコホモロジー環(2) 正則な半単純ヘッセンバーグ多様体のコホモロジー環のワイル群不変部分環(3) ワイル配置の部分配置であるイデアル配置の対数的導分加群を用いて定まる剰余環\par 本研究は阿部拓郎氏、枡田幹也氏、村井聡氏、佐藤敬志氏との共同研究である。
10:45--11:45

ルート系のLinial配置と特性準多項式
13:30--14:30

ワイル群のミヌスクル元からの挿入／削除の拡張
符号理論の１分野として挿入／削除誤りを訂正する符号の研究がある。この研究は1960年代のLevenshteinの仕事に端をなすが、難しい分野であり現在もゆっくりと研究が進展している。　本研究ではそもそもの現象である「挿入」とは何かをワイル群、特にB型のミヌスクル元の視点から問い直す。さらに、挿入の逆の操作と言える「削除」や、Levenshteinの仕事、それらに付随する研究を再構成していく。　一方で、A型の「挿入」を定義・導入する。これを本研究ではバランス隣接挿入と呼び、その逆の操作である「バランス隣接削除」、Levenshteinの仕事のA型版などを創出していく。　なお、本発表は、SITA2016、ISIT2017などの成果を数学的に再整理したものを軸としつつ、新たな結果を加えたものである。
14:45--15:45
Maria GORELIK (The Weizmann Institute of Science)
On Duflo-Serganova functor for affine Lie superalgebras
In this talk (based on a joint work with V. Serganova) I will discuss applications of Dulfo-Serganova functor to affine Lie superalgebras and vertex algebras.
16:00--17:00

Enhanced adjoint action and its quotients
We studied a two-sided enhanced adjoint action of the general linear group $G = \mathrm {GL}_n(\mathbb {C})$ on $$W = (\mathbb {C}^n)^{\oplus p} \oplus (\mathbb {C}^{\ast n})^{\oplus q} \oplus \mathrm {M}_n = \mathrm {M}_{n,p} \oplus \mathrm {M}_{q, n} \oplus \mathrm {M}_n ,$$ a product of its Lie algebra $\mathfrak {gl}_n(\mathbb {C}) = \mathrm {M}_n$ and a vector space consisting of several copies of defining representations and its duals. The invariant ring of $W$ is studied by Le Bruyn and Procesi and also Minoru Itoh. \par We determined regular semisimple orbits (i.e., closed orbits of maximal dimension) and the structure of enhanced null cone, including its irreducible components and their dimensions. However, entire strucure of the orbit space still remains open. \par In this talk, we will focus on the case $p = q = 1$, where the action of $G$ is coregular. \par The talk is based on \texttt {arXiv:1703.08641}.

#### 10月13日 (金)

9:30--10:30

Graded characters of generalized Weyl modules and Demazure submodules of level-zero extremal weight modules
Feigin-Makedonskyi introduced generalized Weyl modules $W_{w \lambda }$, $w \in W$, over the positive part of the affine Lie algebra in the study of the specialization of nonsymmetric Macdonald polynomials at $t=\infty$, where $W$ is a finite Weyl group and $\lambda$ is a dominant weight. In this talk, for an arbitrary $w \in W$, we explain the equality between the graded character of the generalized Weyl module $W_{w \lambda }$ and that of a specific quotient $V_{w}^{-}(\lambda ) / X_{w}^{-}(\lambda )$ of the Demazure submodule $V_{w}^{-}(\lambda )$ of the extremal weight module $V(\lambda )$ over the quantum affine algebra. Also, as an application, we establish a kind of expansion formula for the graded character of $V_{w}^{-}(\lambda +\mu ) / X_{w}^{-}(\lambda +\mu )$ for $w \in W$, which is described in terms of the graded characters of $V_{v}^{-}(\lambda ) / X_{v}^{-}(\lambda )$ and $V_{u}^{-}(\mu ) / X_{u}^{-}(\mu )$, $u,v \in W$, where $\lambda$ and $\mu$ are dominant weights.
10:45--11:45

Riemann ゼータ関数の一般化の一つに（Euler-Zagier 型）多重ゼータ関数と呼ばれるものがある。これは数論に限らず、数学や数理物理学の様々な分野と関連があることが分かっており、現在非常に活発に研究されている。本講演では、この多重ゼータ関数を Schur 関数の視点から拡張した"Schur 多重ゼータ関数" なるものを導入し、それについての基本的な性質や関係式について紹介する。なお本講演の内容は、上智大学の中筋麻貴氏とチュラーロンコーン大学 のOuamporn Phuksuwan 氏との共同研究(arXiv:1704.08511) に基づくものである。\par (念のために英語版も以下に書きます。) The multiple zeta functions (of Euler-Zagier type) are generalizations of the Riemann zeta function. Now, they are actively studied by many researchers because it is known that they appear in various fields of mathematics and mathematical physics. In this talk, as an extension of the multiple zeta functions, we will introduce a Schur multiple zeta function form the Schur function point of view and explain some fundamental properties of them and relations among them. This is based on a joint work (arXiv:1704.08511) with Maki Nakasuji (Sophia University) and Ouamporn Phuksuwan (Chulalongkorn University).

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