## About the seminar

Many interesting phenomena in physics can be described by algebriac objects that have combinatorial interpretations. Some of these include quantum groups and crystal bases, quiver varieties, and solvable lattice models. This seminar series focuses on studying these interactions.

We meet fortnightly on Wednesdays at 10 AM (Japan Standard Time). All of our talks will be broadcast via Zoom with a hybrid setting at OCAMI when possible. The organizers are:

- Travis Scrimshaw (tcscrims {at} gmail.com)
- Hideya Watanabe (hideya {at} kurims.kyoto-u.ac.jp)

If you wish to be added to the mailing list, please contact Travis Scrimshaw (tcscrims {at} gmail.com).

### Upcoming talks

Date | Speaker | Title | Abstract |
---|---|---|---|

March 6th, 2024 | Theo Assiotis | Dynamics in interlacing arrays, conditioned walks and the Aztec diamond | I will discuss certain dynamics of interacting particles in interlacing arrays with inhomogeneous, in space and time, jump probabilities and their relations to conditioned random walks and random tilings of the Aztec diamond. |

### Previous talks

Date | Speaker | Title | Abstract |
---|---|---|---|

October 18th, 2023 | Matthew Nicoletti | Colored interacting particle systems on the ring: Stationary measures from Yang–Baxter equation | Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1)~the Asymmetric Simple Exclusion Process (mASEP); (2)~the \(q\)-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the \(q\)-Boson particle system; (3)~the \(q\)-deformed Pushing Totally Asymmetric Simple Exclusion Process (\(q\)-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new "queue vertex models" on the cylinder. The stationarity property is a direct consequence of the Yang--Baxter equation. This is joint work with A. Aggarwal and L. Petrov. |

July 12th, 2023 | Sin-Myung 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 finite-dimensional 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 finite-dimensional 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 Jae-Hoon Kwon and Masato Okado (arXiv:2304.06215). |

March 24, 2023 | Yuki Kanakubo | 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 \(A^{(1)}_{n-1}\), \(D^{(2)}_n\) and \(\iota\) satisfies a condition called adapted. We will also provide a combinatorial description of inequalities in the \(C^{(1)}_{n-1}\), \(A^{(2)}_{2n-2}\)-cases. |

December 2, 2022 | Anthony Lazzeroni | Powersum bases in Quasisymmetric functions and Quasisymmetric functions in Non-Commuting 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 non-commuting variables by introducing fillings with disjoint sets. |

November 4, 2022 | Shuichiro Matsuike | New solutions to the tetrahedron equation associated with quantized six-vertex models | Tetrahedron equation is a key to integrability for 3-dimensional lattice models, as the Yang–Baxter equation is for 2-dimension. Compared with the Yang–Baxter equation, the tetrahedron equation is much less studied. On the other hand, there are some interesting connections of the 3-dimensional integrability with the representation theory of quantized coordinate rings, reductions to the 2-dimensional 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 six-vertex 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 2-dimensional and 3-dimensional integrable lattice models, in addition to the results. |

September 9, 2022 | Ryo Ohkawa | Wall-crossing for vortex partition function and handsaw quiver varierty | We investigate the partition function determined by integration over the handsaw quiver variety of type \(A_1\). This gives another proof of the transformation formula for multiple hypergeometric series, including the rational version of the Kajiwara transformation formula. This proof is based on a geometric method developed by Nakajima–Yoshioka based on Mochizuki's over-the-wall formula. This time, I will not touch on this proof, but will explain the derivation of the explicit formula and its relation to multiple hypergeometric series. This research is based on joint research with Yutaka Yoshida. |

August 12, 2022 | 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. |

July 15, 2022 | 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 non-standard deformation \(U_q'(\mathfrak{so}_m)\) on the exterior algebra \(\bigwedge(\mathbb{C}^{nm}) \cong S^{\otimes m}\) and obtain a multiplicity-free 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 co-ideal subalgebra of \(U_q(\mathfrak{gl}_m)\), so our construction fits in a see-saw 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. |

July 1, 2022 | Andrea Appel | Schur–Weyl dualities for quantum affine symmetric pairs | In the work of Kang, Kashiwara, Kim, and Oh, the Schur-Weyl 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 K-matrix of the quantum symmetric pair. |

June 24, 2022 Unusual time: 9 AM JST |
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 three-fold: obtaining finite-dimensional algebras related to the braid group, providing solutions of the Yang–Baxter equation, and generalising the Schur–Weyl duality. The braid-like 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 representation-theoretic 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. |

June 17, 2022 | Il-Seung Jang | Unipotent quantum coordinate ring and prefundamental representations | A prefundamental representation is an infinite-dimensional 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 representation-theoretical 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 Jae-Hoon 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 Jae-Hoon Kwon and Euiyong Park (arXiv:2103.05894). |

May 20, 2022 | Jae-Hoon 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 finite-dimensional 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 Sin-Myung Lee (arXiv:2203.12862) |

Apr. 22, 2022 | Yaolong Shen | The q-Brauer algebra and i-Schur duality | Brauer introduced the Brauer algebra, and established the double centralizer property
between it and the orthogonal group or symplectic group. Wenzl defined a q-deformation
of the Brauer algebra which contains the type A Hecke algebra as a natural subalgebra.
It is well known that Jimbo-Schur duality relates Hecke algebras and quantum groups of
type A. In recent years, Bao and Wang have formulated a q-Schur 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 i-Schur duality between them and the q-Brauer algebra. We also develop a natural bar involution and construct a Kazhdan-Lusztig type canonical basis on the q-Brauer algebra. This is joint work with Weideng Cui. |

Apr. 8, 2022 | Masahide Manabe | 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. |

Mar. 25, 2022 | Liron Speyer | Schurian-infinite 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 Schurian-finite if there are only finitely many isomorphism classes of Schurian \(A\)-modules, and Schurian-infinite otherwise. I will present recent joint work with Susumu Ariki in which we determined that many blocks of type \(A\) Hecke algebras are Schurian-infinite. 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. |

Mar. 11, 2022 | Weinan Zhang | Braid group symmetries on i-quantum groups | Introduced by Lusztig in the early 1990s, the braid group symmetries constitute an essential part in the theory of quantum groups. The i-quantum 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 i-quantum 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. |

Feb. 25, 2022 | Takeshi Ikeda | Combinatorial description of the K-homology 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 non-commutative 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. |

Feb. 9, 2022 5 PM |
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. |

Jan. 28, 2022 | 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 quasi-split \(i\)-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. |