\(
\def\tab#1#2#3#4#5{
\begin{array}{ll}
\text{#1} & \begin{array}{ll} \text{#2} & \text{(#3)} \end{array} \\
& \begin{array}{l} \text{ #4 } \\ \text{ #5 }\end{array} \\
\end{array}
}
\)
RIMS 共同研究（公開型）
合流する組み合わせ論と表現論
Confluence of combinatorics and representation theory
京都大学数理解析研究所の共同事業の一つとして, 下記のように研究集会を開催しますので,
ご案内申し上げます。
研究代表者 仲田研登（岡山大学）
記
日程：2025年10月21日(火) 14：00 ～ 10月24日(金) 12：00
会場：京都大学数理解析研究所4 階420 号室
(京都市左京区北白川追分町, 市バス「京大農学部前」または「北白川」下車）
プログラム
pdf版
pdf版(改訂版)
10月21日(火)
\(
\tab{14:00～14:50}{小林 雅人}{神奈川大学}{
Gessel, Assaf-Searlesの準対称関数に関する定理の拡張}{
An extension of theorems on quasi-symmetric functions by Gessel and Assaf-Searles
}
\)
abstract
The “fundamental” theorem of quasi-symmetric functions is
Gessel's result (1984) on decomposing Schur function to the sum of
quasi-symmetric functions up to descent of standard Young tableaux.
More recently, Assaf-Searls (2017) improved it to extract only nonzero
terms by introducing quasi-Yamanouchi tableaux.
We naturally extend these results for semistandard oscillating tableaux (SSOTs),
a brand new object,
as Lee (2025) introduced to prove a type \(C\) crystal structure for King tableaux.
We will moreover show that the generating function of SSOTs is symmetric as
we expect.
This is a joint work with Tomoo Matsumura and Shogo Sugimoto (International
Christian University).
\(
\tab{15:05～15:55}{廣田 竣介}{京都大学}{
奇Vermaの定理}{
Odd Verma's theorem
}
\)
abstract
半単純Lie代数の分類は古典的root系がよく説明するが、
その超代数版は、本質的に異なる基底の取り方を許すようなroot系というべきWeylgroupoidがよく説明するものだと考える。
Weyl groupoidはスーパーなる概念と定義からは関係せず、
frieze patternを一般化する純粋な組み合せ的対象である。
古典的なroot系/Weyl群の組み合せ的性質は、表現論に様々な形で現れていたので、
Weyl groupoidの場合どうかが気になるが、
Lie超代数の表現論においてそれは奇鏡映を気にすることに対応する。
本講演ではそのような方向性での最も基本的な結果の一つと考えられる奇(鏡映版)Vermaの定理とでもいうべき結果について紹介する。
これは奇鏡映の交換法則という性質の帰結であり、それはWeylgroupoidの交換法則なるものから導けることが重要である。
\(
\tab{16:10～17:00}{澁川 陽一}{北海道大学}{
箙上のヤン・バクスター方程式のガーサイド理論}{
Garside theory of the Yang-Baxter equation on quivers
}
\)
abstract
圏の表示を用いて箙上のヤン・バクスター方程式の解を構成する．
逆に，箙上のヤン・バクスター方程式の解が定める商圏の性質について，
ガーサイド理論的な観点から説明する．
10月22日(水)
\(
\tab{10:00～10:50}{藤田 遼}{京都大学数理解析研究所}{
Applications of freezing operator to the representation theory of quantum loop algebras}{
}
\)
abstract
In the last decade, the representation theory of quantum loop algebras
has been intensively studied from the viewpoint of cluster theory.
Recently, Fan Qin introduced a certain algebraic operator, called the
freezing, relating good elements in different cluster algebras. In
this talk, we discuss applications of the freezing operator to the
representation theory of quantum loop algebras, including a proof of
the Hernandez conjecture (an analog of Kazhdan--Lusztig conjecture)
for type B and C. This is based on a joint work with Fan Qin.
\(
\tab{11:05～11:55}{Alessandro Contu}{京都大学 数理解析研究所}{
Solution of a problem in monoidal categorification by additive categorification}{
}
\)
abstract
In 2021, Kashiwara--Kim--Oh--Park constructed cluster algebra
structures on the Grothendieck rings of certain monoidal subcategories of
the category of finite-dimensional representations of a quantum loop
algebra, generalizing Hernandez--Leclerc's pioneering work from 2010. They
stated the problem of finding explicit quivers for the seeds they used. We
provide a solution by using Palu’s generalized mutation rule applied to the
cluster categories associated with certain algebras of global dimension at
most \(2\), for example tensor products of path algebras of
representation-finite quivers. Our method is based on (and contributes to)
the bridge, provided by cluster combinatorics, between the representation
theory of quantum groups and that of quivers with relations.
\(
\tab{14:00～14:50}{金久保 有輝}{茨城大学}{
Inequalities defining polyhedral realizations and monomial realizations of crystal bases}{
}
\)
abstract
Crystal bases are powerful tools to study representations of Lie algebras and quantum groups.
We can get several essential information of integrable highest weight representations or Verma modules from them.
Crystal bases have a bunch of descriptions via combinatorial objects, which enable us to combinatorially study them.
The polyhedral realizations invented by Nakashima-Zelevinsky are such descriptions in terms of the set of integer points of a convex cone,
which coincides with the string cone when the associated Lie algebra is finite dimensional simple.
It is a fundamental and natural problem to find an explicit form of this convex cone.
The monomial realizations introduced by Kashiwara and Nakajima are other expressions of crystal bases as Laurent monomials in double indexed variables.
In this talk,
we give a conjecture that the inequalities defining the cone of polyhedral realizations can be expressed in terms of monomial realizations of fundamental representations,
which is true when the Lie algebra is classical type or classical affine type.
\(
\tab{15:05～15:55}{Duc-Khanh Nguyen}{沖縄科学技術大学院大学, OIST}{
Branching rule on winding subalgebras of affine Kac-Moody algebras}{
}
\)
abstract
In this work, by using the Lakshmibai-Seshadri paths,
we give the branching rule for representations of affine Kac-Moody algebras to their winding subalgebras.
As a corollary, we can describe branching multiplicities in the language of paths.
An analog of Steinberg's formula for branching multiplicities is also given.
\(
\tab{16:10～17:00}{中島 啓}{東京大学 カブリ数物連携宇宙研究機構}{
Involution on quiver varieties and quantum symmetric pairs}{
}
\)
abstract
Seven years ago, Yiqiang Li considered equivariant cohomology of fixed
point sets of involutions on quiver varieties, -- \(\sigma\)-quiver varieties,
and constructed representations of coideal subalgebras of Yangian, i.e.,
twisted Yangian. We remove the unnecessary assumption which Li imposed, and
calculate the twisted Yangian explicitly in some examples.
10月23日(木)
\(
\tab{10:00～10:50}{内海 凌}{大阪大学}{
有限群が作用する超平面配置とその特性準多項式}{
Hyperplane arrangements with finite group actions and characteristic quasi-polynomials}{
}
\)
abstract
整数係数で定義される超平面配置に対して，
対応する \( \mod q \) 配置の補空間の数え上げ関数として特性準多項式が定義される。
これは超平面配置の重要な不変量である特性多項式を含んだものである．
また，エルハート準多項式の \( \mod q \) 版といわれることもある。
一方，エルハート理論では，その同変理論が導入されており，近年盛んに研究されている。
本講演では，有限群が作用する超平面配置に対して，
同変版の特性準多項式を，対応する \( \mod q \) 配置の補空間に関する置換指標として導入し，
同変版エルハート準多項式との関連を述べる。
\(
\tab{11:05～11:55}{Hau-Wen Huang}{National Central University, Taiwan}{
A skew group ring of $\mathbb{Z}/2\mathbb{Z}$ over $U(\mathfrak{sl}_2)$, Leonard triples and odd graphs}{
}
\)
abstract
The universal Bannai--Ito algebra \(\mathfrak{BI}\) is an algebra over \(\mathbb{C}\) with generators \(X,Y,Z\) and the relations assert that each of
\( \{X,Y\}-Z , \{Y,Z\}-X, \{Z,X\}-Y \)
is central in \(\mathfrak{BI}\).
The universal enveloping algebra \(U(\mathfrak{sl}_2)\) of \(\mathfrak{sl}_2\) is an algebra over \(\mathbb{C}\) generated by \(E,F,H\) subject to the relations
\([H,E]=2E, [H,F]=-2F, [E,F]=H \).
Merging with the algebra involution of \(U(\mathfrak{sl}_2)\) given by \( (E,F,H)\mapsto (F,E,-H) \),
this produces a skew group ring \( U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}} \) of \( \mathbb{Z}/2\mathbb{Z} \) over \( U(\mathfrak{sl}_2) \).
The skew group ring \( U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}} \) is a Hopf algebra.
Let \( \Delta \) denote the comultiplication of \( U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}} \).
Let \(V\) denote a \( U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}}^{\otimes 2} \)-module.
For any \( \theta\in \mathbb{C} \) let \(V(\theta)\) consist of all \(v\in V\) with \(\Delta(H)v=\theta v\).
In this talk, I will display a \(\mathfrak{BI}\)-module structure on \(V(1)\).
Suppose that \(V\) is a finite-dimensional irreducible \(U(\mathfrak{sl}_2)_{\mathbb{Z}/2\mathbb{Z}}^{\otimes 2}\)-module and that \(V(1)\) is nonzero.
Then the \(\mathfrak{BI}\)-module \(V(1)\) is irreducible and \(X,Y,Z\) act on \(V(1)\) as a Leonard triple.
Let \(d\geq 1\) be an integer.
Fix a vertex \(x_0\) of the odd graph \(O_{d+1}\).
Let \(\mathbf{T}(x_0)\) denote the Terwilliger algebra of \(O_{d+1}\) with respect to \(x_0\).
As an application, there exists an algebra homomorphism \(\mathfrak{BI} \to \mathbf{T}(x_0)\) and \(X,Y,Z\) act on each irreducible \(\mathbf{T}(x_0)\)-module as a Leonard triple.
This is a joint work with Dr. Chin-Yen Lee.
\(
\tab{14:00～14:50}{大久保 勇輔}{摂南大学}{
$N=1$ 超対称Virasoro代数と自由fermion代数の直和の$q$-変形}{
A $q$-deformation of the direct sum of the $N=1$ super Virasoro algebra and the free fermion algebra
}
\)
abstract
\(q\)-変形Virasoro代数は，Macdonald多項式や5次元ゲージ理論（\(q\)-変形版のAGT対応）などに深く関わる重要な量子代数であり，
その発見以来，多くの研究が進められてきた．
この代数は，量子トロイダル\(\mathfrak{gl}_1\)代数（アフィン量子群のある種の一般化）の自由場表示から導出できることが知られている．
本講演では，この導出の枠組みを\(\mathfrak{gl}_2\)型の量子トロイダル代数に適用することで得られる新しい代数の性質について紹介する．
この場合に現れる代数は，\(N=1\)超対称Virasoro代数の\(q\)-変形（ただし1成分の自由fermion代数の寄与を含むもの）と解釈できる．
さらに，この代数は2つのスクリーニング作用素をもち，閉じた2次関係式を満たす．
これを用いた最高ウェイト表現の解析から，Kac行列式の因子化も予想される．
\(
\tab{15:05～15:55}{渡邉 英也}{立教大学}{
Bereleの行挿入と量子対称対}{
Berele row-insertion and quantum symmetric pairs
}
\)
abstract
Bereleの行挿入は、Schenstedの行挿入の斜交群類似であり、斜交盤と正整数の組から新しい斜
交盤を生成するアルゴリズムである。
本講演では、量子対称対を用いてこのアルゴリズムの表現論的解釈を与える。
応用として、斜交盤を用いたRobinson--Schensted--Knuth型対応という組合せ論的な結果の表現
論的解釈や、斜交群の既約表現と一般線形群の既約表現のテンソル積を斜交群の表現として既約
分解するという表現論の問題に対する組合せ論的解法が得られることを説明する。
\(
\tab{16:10～17:00}{岡田 聡一}{名古屋大学}{
Refinements of bounded Littlewood identities}{
}
\)
abstract
The Schur-Littlewood identity is an identity equating a sum of all Schur functions with an infinite product,
while bounded Littlewood identities concern the sums of Schur functions over partitions with a bounded number of rows or columns.
In this talk, we give refinements of bounded Littlewood identities with respect to the number of odd-length columns.
As applications, we present several formulas for the number of standard Young tableaux with restricted shapes.
10月24日(金)
\(
\tab{10:00～10:50}{松浦 滉}{上智大学}{
圏$\widetilde{R\hbox{-gmod}}$におけるunit object の特徴づけ}{
}
\)
abstract
Let \(R\) be the quiver Hecke algebra \(\mathfrak{g}\),
\(R\hbox{-gmod}\) be the category of graded finite-dimensional \(R\)-modules and \(\mathscr{C}_w\) be the subcategory of \(R\hbox{-gmod}\) associated with a Weyl group element \(w\).
The localization of these categories,
\(\widetilde{R\hbox{-gmod}}\) and \(\widetilde{\mathscr{C}_w}\),
have been defined.
Recently, Kashiwara and Nakashima showed \(\mathrm{Irr}( \widetilde{\mathscr{C}_w})\),
the set of equivalence classes of simple objects in \(\widetilde{\mathscr{C}_w}\),
possesses a crystal structure,
and is isomorphic to the so-called cellular crystal \(\mathbb{B}_{\mathbf{i}}\).
This isomorphism induces a function \(\varepsilon_i^*\) on \(\mathbb{B}_{\mathbf{i}}\).
In this talk, I will present an explicit formula for \(\varepsilon_i^*\) and then,
using this formula, provide a characterization of the unit object of \(\widetilde{R\hbox{-gmod}}\).
This is joint work with T. Nakashima.
\(
\tab{11:05～11:55}{中桐 正人}{東京大学}{
Worley-Sagan挿入とHaimanの混合挿入の双対性を保つ超八面体群への拡張}{
Duality-Preserving Extensions of the Worley-Sagan Insertion and Haiman's Mixed Insertion for the Hyperoctahedral Group
}
\)
abstract
The Worley-Sagan insertion and Haiman’s mixed insertion are
insertion algorithms on shifted Young tableaux; each yields a
Robinson-Schensted-type correspondence between the symmetric group \(S_n\)
and pairs of shifted Young tableaux of the same shape with \(n\) cells. It is
well known that these two insertions are dual: if a permutation \(\pi\)
corresponds to \((P, Q)\) via the Worley-Sagan insertion, then \(\pi^{-1}\)
corresponds to \((Q, P)\) via Haiman’s mixed insertion. In this talk we
present extensions of these insertions that preserve this duality. Our
extensions map signed permutations to pairs of shifted tableaux. Note that
our extension of the Worley-Sagan insertion differs from restricting
Sagan’s “Knuth version” to signed permutations. To prove duality for our
extensions, we embed them into the doubly mixed insertion introduced by
Shimozono and White on unshifted tableaux via a doubling construction for
shifted tableaux, and then appeal to the self-duality of the doubly mixed
insertion established by Shimozono and White.