セミナーについて
組合せ理論での解釈がある代数的な物で説明できる面白い現象は物理学で多くあります。 例えば、quantum groups と crystal bases、quiver varieties、 solvable lattice modelsです。 このセミナーの目的は、これらの関係を研究することです
隔週金曜日の午後3時（日本時間）に、Zoom または大阪市立大学数学研究所で行います。 セミナーの世話人は以下の通りです：
 S
CRIMSHAW Travis (tcscrims {at} gmail.com) 渡邉英也 (hideya {at} kurims.kyotou.ac.jp)
メーリングリストに追加するには、 Travis Scrimshaw (tcscrims {at} gmail.com)にご連絡ください。
今後の講演
日付  講演者  題名  要約 

July 12th, 2023  SinMyung Lee  Oscillator representations of quantum affine orthosymplectic superalgebras  We introduce a category of \(q\)oscillator representations over the quantum affine superalgebras of type D and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible \(q\)oscillator representations of affine type \(X\) and the finitedimensional irreducible representations of affine type \(Y\) for \((X,Y) = (C,D), (D,C)\) under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finitedimensional representations of classical Lie algebras arising from the Howe’s reductive dual pairs \((g,G)\), where \(g = sp_{2n}, so_{2n}\) and \(G = O_l, Sp_{2l}\). This talk is based on a joint work with JaeHoon Kwon and Masato Okado (arXiv:2304.06215). 
以前の講演
日付  講演者  題名  要約 

2023年3月24日  Polyhedral realizations and extended Young diagrams of several classical affine types  The crystal bases are powerful tools for studying the representation theory of quantized enveloping algebras. By realizing crystal bases as combinatorial objects, one can reveal skeleton structures of representations. Nakashima and Zelevinsky invented 'polyhedral realizations', which are realizations of crystal bases \(B(\infty)\) and \(B(\lambda)\) as integer points in some polyhedral convex cones or polytopes. It is a natural problem to find an explicit form of inequalities that define the polyhedral convex cones and polytopes. To construct the polyhedral realization, we need to fix an infinite sequence \(\iota\) of indices. In this talk, we will give an explicit form of inequalities in terms of extended Young diagrams in the case the associated Lie algebra is classical affine type of \(A^{(1)}_{n1}\), \(D^{(2)}_n\) and \(\iota\) satisfies a condition called adapted. We will also provide a combinatorial description of inequalities in the \(C^{(1)}_{n1}\), \(A^{(2)}_{2n2}\)cases.  
2022年12月2日  Anthony Lazzeroni  Powersum bases in Quasisymmetric functions and Quasisymmetric functions in NonCommuting variables  We introduce a new P basis for the Hopf algebra of quasisymmetric functions that refine the symmetric powersum basis. Its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. The product and coproduct are then extended to matrix fillings thereby defining a Hopf algebra of matrix fillings We lift our quasisymmetric powersum P basis to the Hopf algebra of quasisymmetric functions in noncommuting variables by introducing fillings with disjoint sets. 
2022年11月4日  New solutions to the tetrahedron equation associated with quantized sixvertex models  Tetrahedron equation is a key to integrability for 3dimensional lattice models, as the Yang–Baxter equation is for 2dimension. Compared with the Yang–Baxter equation, the tetrahedron equation is much less studied. On the other hand, there are some interesting connections of the 3dimensional integrability with the representation theory of quantized coordinate rings, reductions to the 2dimensional cases, and so on. In our recent paper (A. Kuniba, S. M. and A. Yoneyama, arXiv:2208.10258), we considered the tetrahedron equation of the form \(RLLL=LLLR\), which is called a quantized Yang–Baxter equation. The \(L\) operators here are built from \(q\)Weyl algebra and can be regarded as quantized sixvertex models. The solutions \(R\) are obtained explicitly in many cases, whose elements are factorized or expressed with a terminating \(q\)hepergeometric series. This talk will include an introduction to 2dimensional and 3dimensional integrable lattice models, in addition to the results. 

2022年9月9日 

Wallcrossing for vortex partition function and handsaw quiver varierty  \(A_1\)型handsaw quiver variety上の積分により定まる分配関数を調べる. これにより梶原変換公式の有理版をふくむ多重超幾何級数の変換公式に別証明を与えた. この証明は望月氏の壁越え公式に基づく中島吉岡の仕事を発展させた幾何学的な方法による. 今回はこの証明には触れず、明示公式の導出や多重超幾何級数との関連について説明する. 本研究は、吉田裕氏との共同研究に基づく. 
2022年8月12日  Jihyeug Jang  A combinatorial model for the transition matrix between the Specht and web bases  We introduce a new class of permutations, called web permutations. Using these permutations, we provide a combinatorial interpretation for entries of the transition matrix between the Specht and web bases of the irreducible \(\mathfrak{S}_{2n}\)representation indexed by \((n,n)\), which answers Rhoades's question. Furthermore, we study enumerative properties of these permutations. 
2022年7月15日  Guillermo Antonio "Willie" Aboumrad Sidaoui  Explicit solutions of the Yang–Baxter equation via skew Howe duality and \(F\)matrices  We construct matrix solutions to the Yang–Baxter Equation (YBE) associated to the spinor object in the fusion ring \(O(2n)_2\) in two ways. Let \(S = S_+ \oplus S_\) denote the spinor module for \(U_q(\mathfrak{o}_{2n})\) and by slight abuse of notation its image in the fusion quotient \(O(2n)_2\). On one hand, first we define commuting actions of \(U_q(\mathfrak{o}_{2n}) = U_q(\mathfrak{so}_{2n}) \rtimes \mathbb{Z}_2\) and the nonstandard deformation \(U_q'(\mathfrak{so}_m)\) on the exterior algebra \(\bigwedge(\mathbb{C}^{nm}) \cong S^{\otimes m}\) and obtain a multiplicityfree decomposition. Then we construct solutions of the YBE as certain polynomials in the generators of \(U_q'(\mathfrak{so}_m)\). Our construction factors the \(U_q(\mathfrak{o}_{2n}) \otimes U_q'(\mathfrak{so}_m)\)action through the quantum Clifford algebra \(Cl_q(nm)\) and carries over to the fusion quotient. In addition, the general linear quantum group \(U_q(\mathfrak{gl}_n)\) is a subalgebra of \(U_q(\mathfrak{so}_{2n})\) and \(U_q'(\mathfrak{so}_m)\) is a coideal subalgebra of \(U_q(\mathfrak{gl}_m)\), so our construction fits in a seesaw extending a \(U_q(\mathfrak{gl}_n)\otimes U_q(\mathfrak{gl}_m)\) skew duality result. On the other hand, we construct solutions of the YBE by considering the structure of the fusion ring \(O(2n)_2\). In particular, first we solve the Pentagon equations to obtain a set of \(F\)matrices for \(O(2n)_2\). Then for each \(X \in O(2n)_2\) we choose a fusion basis for \(\text{Hom}(S^{\otimes m}, X)\) and apply an appropriate sequences \(F\) and \(R\)moves. 
2022年7月1日  Andrea Appel  Schur–Weyl dualities for quantum affine symmetric pairs  In the work of Kang, Kashiwara, Kim, and Oh, the SchurWeyl duality between quantum affine algebras and affine Hecke algebras is extended to certain Khovanov–Lauda–Rouquier (KLR) algebras, whose defining combinatorial datum is given by the poles of the normalised R–matrix on a set of representations. In this talk, I will review their construction and introduce a ‘’boundary'' analogue, consisting of a Schur–Weyl duality between a quantum symmetric pair of affine type and a modified KLR algebras arising from a (framed) quiver with a contravariant involution. With respect to the Kang–Kashiwara–Kim–Oh construction, the extra combinatorial datum we take into account is given by the poles of the normalised Kmatrix of the quantum symmetric pair. 
2022年6月24日 午前9時 
Loic Poulain d'Andecy  Fused Hecke algebra and quantum Schur–Weyl duality  This talk will be about the algebra of fused permutations and the fused Hecke algebra. The motivation for considering these algebras is threefold: obtaining finitedimensional algebras related to the braid group, providing solutions of the Yang–Baxter equation, and generalising the Schur–Weyl duality. The braidlike diagrammatic description of the fused Hecke algebra will be presented together with its combinatorial shadow of fused permutations. An explicit construction of solutions of the Yang–Baxter equation inside these algebras will be given. Finally, the meaning of these algebras in the context of the Schur–Weyl duality will be explained: they provide a description of the centralisers of tensor products of symmetric powers representations of quantum groups. The representation theory of the fused Hecke algebra will be explained and, from this, the centralisers can be constructed as quotients, both with a representationtheoretic and an algebraic/diagrammatic description. These constructions can be seen in particular as analogues for higher spins of a construction of the Temperley–Lieb algebras from the Hecke algebras. This is a joint work with Nicolas Crampe. 
2022年6月17日  IlSeung Jang  Unipotent quantum coordinate ring and prefundamental representations  A prefundamental representation is an infinitedimensional simple module over the Borel subalgebra of quantum loop algebra (of untwisted type), which was introduced by Hernandez and Jimbo (arXiv:1104.1891) to give a representationtheoretical interpretation of the stability for Kirillov–Reshetikhin (KR for short) modules. This talk will explain a new realization of the prefundamental representations associated with minuscule nodes (for types A and D) by using the unipotent quantum coordinate ring associated with the translation by a fundamental weight. It was motivated by the works of JaeHoon Kwon (arXiv:1110.2629, arXiv:1606.06804), in which certain KR crystals were constructed from the crystal of the unipotent quantum coordinate ring. This is joint work with JaeHoon Kwon and Euiyong Park (arXiv:2103.05894). 
2022年５月20日  JaeHoon Kwon  Oscillator representations of quantum affine algebras of type A  In this talk, we introduce a category of oscillator representations of the quantum affine algebra for \(gl_n\). This can be viewed as a quantum affine analog of the semisimple tensor category generated by unitarizable highest weight representations of \(gl_{u+v}\) (\(n=u+v\)) appearing in the \((gl_{u+v},gl_\ell)\)duality on a bosonic Fock space. We show that it has a family of irreducible representations, which naturally correspond to finitedimensional irreducible representations of the quantum affine algebra of type A. This is done by considering oscillator representations of quantum affine superalgebras of type A, which interpolates these two categories via certain monoidal functors preserving the R matrices.This is joint work with SinMyung Lee (arXiv:2203.12862) 
2022年4月22日  Yaolong Shen  The qBrauer algebra and iSchur duality  Brauer introduced the Brauer algebra, and established the double centralizer property
between it and the orthogonal group or symplectic group. Wenzl defined a qdeformation
of the Brauer algebra which contains the type A Hecke algebra as a natural subalgebra.
It is well known that JimboSchur duality relates Hecke algebras and quantum groups of
type A. In recent years, Bao and Wang have formulated a qSchur duality between a type B Hecke algebra and an iquantum group arising from quantum symmetric pairs. In this talk we focus on iquantum groups which specialize to the orthogonal or symplectic Lie algebra at q=1 and introduce an iSchur duality between them and the qBrauer algebra. We also develop a natural bar involution and construct a KazhdanLusztig type canonical basis on the qBrauer algebra. This is joint work with Weideng Cui. 
2022年4月8日 

2D CFT characters from 4D \(U(N)\) instanton counting on \(\mathbb{C}^2/\mathbb{Z}_n\)  As a type of the Alday–Gaiotto–Tachikawa correspondence, we explicitly show how the \(\widehat{\mathfrak{sl}}(n)_N\) characters and the \(n\)th parafermion \(\mathcal{W}_N\) characters are obtained from 4D \(U(N)\) instanton counting on \(\mathbb{C}^2/\mathbb{Z}_n\) with \(\Omega\)deformation. This is based on arXiv:1912.04407, a joint work with Omar Foda, Nicholas Macleod, and Trevor Welsh, and arXiv:2004.13960. 
2022年3月25日  Liron Speyer  Schurianinfinite blocks of type \(A\) Hecke algebras  For any algebra \(A\) over an algebraically closed field \(\mathbb{F}\), we say that an \(A\)module \(M\) is Schurian if \(\mathrm{End}_A(M) \cong \mathbb{F}\). We say that \(A\) is Schurianfinite if there are only finitely many isomorphism classes of Schurian \(A\)modules, and Schurianinfinite otherwise. I will present recent joint work with Susumu Ariki in which we determined that many blocks of type \(A\) Hecke algebras are Schurianinfinite. A wide variety of techniques were employed in this project — I will give more details on those required for our results concerning the principal blocks. 
2022年3月11日  Weinan Zhang  Braid group symmetries on iquantum groups  Introduced by Lusztig in the early 1990s, the braid group symmetries constitute an essential part in the theory of quantum groups. The iquantum groups are coideal subalgebras of quantum groups arising from quantum symmetric pairs, which can be viewed as natural generalizations of quantum groups. In this talk, I will present our construction of relative braid group symmetries (associated to the relative Weyl group of a symmetric pair) on iquantum groups of arbitrary finite types. These new symmetries inherit most properties of Lusztig's symmetries, including compatible relative braid group actions on modules. This is joint work with Weiqiang Wang. 
2022年2月25日 

Combinatorial description of the Khomology Schubert classes of the affine Grassmannian  Lam, Schilling, and Shimozono constructed a family of inhomogeneous symmetric functions that is identified with a distinguished basis of \(K\)homology of the affine Grassmannian of \(SL_{k+1}\). These functions are called the \(K\)theoretic \(k\)Schur functions and are indexed by the partitions whose largest part is less than or equal to \(k\). Recently, Blasiak, Morse, and Seelinger proved a raising operator formula for the \(K\)\(k\)Schur functions. In fact, they introduced a family of inhomogeneous symmetric functions called the \(K\)theoretic Catalan functions, Katalan functions for short, and proved that the \(K\)theoretic \(k\)Schur functions are a subfamily of Katalan functions. Then also introduced a subfamily of Katalan functions called \(K\)\(k\)Schur "closed" Katalan functions, and conjectured that it is identified with the Schubert basis of the \(K\)homology of the affine Grassmannian. We prove that the conjecture is true by using the theory of noncommutative symmetric functions with affine Weyl group symmetry. I also discuss some connections with the \(K\)theoretic Peterson isomorphism. The talk is based on a joint work with S. Naito and S. Iwao. 
2022年2月9日 午後5時 
Dan Betea  From Gumbel to Tracy–Widom via random (ordinary, plane, and cylindric plane) partitions  We present a few natural (and combinatorial) measures on partitions, plane partitions, and cylindric plane partitions. We show how the extremal statistics of such measures, i.e. the distributions of the largest parts of the respective random objects, interpolate between the Gumbel distribution of classical statistics and the Tracy–Widom GUE distribution of random matrix theory, and do so in more than one way. These laws usually appear in opposite probabilistic contexts: the former distribution (Gumbel) appears universally in the study of extrema of iid random variables, the latter (Tracy–Widom) appears in the extrema of correlated systems (e.g. for the largest eigenvalue of random Hermitian matrices). All statistics also have a last passage percolation interpretation via the Robinson–Schensted–Knuth correspondence. Proofs rely on an interplay between algebraic combinatorics, mathematical physics, and complex analysis. The results are based on joint works with Jérémie Bouttier and Alessandra Occelli. 
2022年1月28日  Weiqiang Wang  Quantum Schur dualities and Kazhdan–Lusztig bases  The standard quantum Schur duality (due to Jimbo) concerns about commuting actions on a tensor space of a quantum group and a Hecke algebra of type A. Several years ago, this was generalized to a duality between a Hecke algebra of type B and a quasisplit \(i\)quantum group arising from quantum symmetric pairs by Bao and myself (and Watanabe in unequal parameters). Both dualities are connected to canonical bases and Kazhdan–Lusztig theory of type ABCD. In this talk, I will explain a unification of both dualities involving more general \(i\)quantum groups, which leads to a new generalization of Kazhdan–Lusztig bases. This is joint work with Yaolong Shen. 