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Organizing Committee
University of Southern California
Universidad Nacional Autónoma de México
York University
Abstract
This workshop encompasses three major aspects of computation within Representation Theory and Algebraic Combinatorics. One concerns the development of efficient algorithms to compute important quantities in order to understand and classify them better. Such problems include structure constants and representation theoretic multiplicities, mutation invariants in cluster algebras, computing dimensions of coinvariant rings, characters of finite-dimensional representations, coefficients of Kazhdan-Lusztig polynomials, web bases etc. This is closely related to understanding what optimality we could expect and in particular the computational complexity aspects of those problems. Their computational complexity class can also be used to understand the existence of combinatorial interpretations, in particular for major structure constants lacking positive formulas like Kronecker and plethysm coefficients. On the other hand, representation theory has seen important applications within computational complexity theory, in the context of Geometric Complexity Theory and Quantum Information Theory.
Last, but not least, we will discuss the collection of “experimental” data, which helps formulate conjectures, find counterexamples and understand the behavior for some of the problems listed above.
Last, but not least, we will discuss the collection of “experimental” data, which helps formulate conjectures, find counterexamples and understand the behavior for some of the problems listed above.
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Workshop Schedule
Monday, November 10, 2025
8:30 - 8:50 AM EST
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EST
Welcome
11th Floor Lecture Hall
9:00 - 9:45 AM EST
Crystal skeletons: Combinatorics and axioms
11th Floor Lecture Hall
Speaker
Anne Schilling, University of California, Davis
Session Chair
Jose Simental Rodriguez, Universidad Nacional Autónoma de México
Abstract
Crystal skeletons were introduced by Maas-Gariépy in 2023 by contracting quasi-crystal components in a crystal graph. On the representation theoretic level, crystal skeletons model the expansion of Schur functions into Gessel's quasisymmetric functions. Motivated by questions of Schur positivity, we provide a combinatorial description of crystal skeletons, and prove many new properties, including a conjecture by Maas-Gariépy that crystal skeletons generalize dual equivalence graphs. We then present a new axiomatic approach to crystal skeletons. We give three versions of the axioms based on GL_n-branching, S_n-branching, and local axioms in analogy to the local Stembridge axioms for crystals based on novel commutation relations. This is based on joint work with Sarah Brauner, Sylvie Corteel and Zajj Daugherty (arXiv:2503.14782).
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Computing Lusztig's q-weight multiplicities beyond type A
11th Floor Lecture Hall
Speaker
Cristian Lenart, University at Albany
Session Chair
Jose Simental Rodriguez, Universidad Nacional Autónoma de México
Abstract
Lusztig's q-weight multiplicities are widely studied objects in the representation theory of semisimple Lie algebras. Those of type A are expressed combinatorially via the Lascoux-Schützenberger charge statistic on semistandard Young tableaux. Generalizing this formula to other classical types has been challenging. I will survey the recent progress, including the work of Choi-Kim-Lee relating the q-weight multiplicities of type C to the energy function on column-shape Kirillov-Reshetikhin (KR) crystals of type B. On another hand, with Adam Schultze, we gave a charge-type formula for the mentioned energy function in terms of the respective Kashiwara-Nakashima columns. This formula was derived from earlier work of mine with Satoshi Naito, Daisuke Sagaki, Anne Schilling, and Mark Shimozono on the quantum alcove model for KR crystals. Thus, by combining these results, we can compute any q-weight multiplicity of type C. I will suggest avenues for extending and refining these results, including the potential use of machine learning.
11:30 AM - 12:15 PM EST
Grobner Crystal Structures
11th Floor Lecture Hall
Speaker
Alexander Yong, University of Illinois at Urbana-Champaign
Session Chair
Jose Simental Rodriguez, Universidad Nacional Autónoma de México
Abstract
We develop a theory of ""bicrystalline ideals"", synthesizing Grobner basis techniques and Kashiwara's crystal theory. This provides a unified algebraic, combinatorial, and computational approach that applies to ideals of interest, old and new. The theory concerns ideals in the coordinate ring of matrices, stable under the action of some Levi group, whose quotients admit standard bases equipped with a crystal structure. We construct an effective algorithm to decide if an ideal is bicrystalline. When the answer is affirmative, we provide a uniform, generalized Littlewood--Richardson rule for computing the multiplicity of irreducible representations either for the quotient or the ideal itself.
This is joint work with Abigail Price (UIUC) and Ada Stelzer (UIUC).
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
Vanishing of Schubert coefficients
11th Floor Lecture Hall
Speaker
Igor Pak, University of California at Los Angeles
Session Chair
Greta Panova, University of Southern California
Abstract
I will give a short overview of complexity problems of vanishing of various structure constants. I will then discuss our recent result that the vanishing of Schubert coefficients can be decided in probabilistic polynomial time. Joint work with Colleen Robichaux.
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EST
Algebraic combinatorics, nonnegative combinatorial interpretations, and computational complexity theory
11th Floor Lecture Hall
Speaker
Christian Ikenmeyer, University of Warwick
Session Chair
Greta Panova, University of Southern California
Abstract
In this talk I will try to shed some light on the notion of a "nonnegative combinatorial interpretation", in particular for characters of the symmetric group, Kronecker coefficients, and plethysm coefficients. This is joint work with Igor Pak, Greta Panova, and Sathyawageeswar Subramanian.
5:00 - 6:30 PM EST
Reception
11th Floor Collaborative Space
Tuesday, November 11, 2025
9:00 - 9:45 AM EST
On a curious variant of Lie_n
11th Floor Lecture Hall
Speaker
Sheila Sundaram, University of Minnesota
Session Chair
Olya Mandelshtam, University of Waterloo
Abstract
This talk will discuss a curious variant of the celebrated representation Lie_n of the symmetric group S_n on the multilinear component of the free Lie algebra on n generators. Introduced by the speaker a few years ago, the variant Lie_n^(2) satisfies the analogue of almost every known property of Lie_n. As one example, the exterior powers of the variant Lie_n^(2) decompose the regular representation of S_n. The classical free Lie algebra counterpart of this result is the 1942 theorem of Robert Thrall which also follows from the earlier Poincaré-Birkhoff-Witt theorem: it is the well-known decomposition of the regular representation given by the symmetrized powers of the representations Lie_n (the higher Lie modules). The talk will survey this and other properties of Lie_n^(2), including some recent developments.
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