Tag - Algebraic combinatorics

Karlee Westrem: A new symmetric function identity with an application to symmetric group character values

7th October, 2024

Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of Σn character sums over a new set called Ev(λ). When investigating the alternating sum of characters for Ev(λ) written in terms of the inner product of Schur functions and power sum symmetric functions, we found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of Σn characters over the set Ev(λ) equals 0.

Duc-Khanh Nguyen: Application of (K-theoretic) Peterson isomorphism

17th September, 2024

The theory of symmetric polynomials plays a key role in Representation Theory, Schubert Calculus, and Algebraic Combinatorics. Fundamental rules like the Pieri, Murnaghan-Nakayama, and Littlewood-Richardson rules describe the decomposition of products of Schubert classes into Schubert classes. We focus on the decomposition of polynomial representatives of Schubert classes in homology and K-homology of the affine Grassmannian of SLn, as well as quantum Schubert classes in quantum cohomology and K-cohomology of the full flag manifold of type A. Specifically, we explore how to use the Peterson isomorphism to connect formulas between homology and quantum cohomology, and between K-homology and quantum K-cohomology, extending techniques from the work of Lam-Shimozono on Schubert classes.

Dominique Manchon: Post-Lie algebras, post-groups and Gavrilov’s K-map

9th September, 2024

Post-Lie algebras appeared in 2007 in algebraic combinatorics, and independently in 2008 in the study of numerical schemes on homogeneous spaces. Gavrilov's K-map is a particular Hopf algebra isomorphism, which can be naturally described in the context of free post-Lie algebras. Post-groups, which are to post-Lie algebras what groups are to Lie algebras, were defined in 2023 by C. Bai, L. Guo, Y. Sheng and R. Tang. Although skew-braces and braided groups are older equivalent notions, their reformulation as post-groups brings crucial new information on their structure. After giving an account of the above-mentioned structures, I shall introduce free post-groups, and describe a group isomorphism which can be seen as an analogon of Gavrilov's K-map for post-groups.

Weiqiang Wang: A new diagrammatic categorical setting for Schur dualities

29th May, 2024

The classical Schur duality is a simple yet powerful concept which relates the representations of the symmetric group and general linear Lie algebra, as well as combinatorics of symmetric functions. This admits a quantum deformation to a duality between a quantum group and Hecke algebra of type A. In this talk, we will describe several new simple diagrammatic (monoidal/quotient) categories, where old and new algebras behind (affine/cyclotomic) Schur duality emerge naturally. Our construction has new combinatorial implications on symmetric functions and RSK correspondence.

Arun Ram: Lusztig varieties and Macdonald polynomials

27th May, 2024

In type A, the Macdonald polynomials and the integral from Macdonald polynomials are related by a plethystic transformation. We interpret this plethystic transformation geometrically as a relationship between nilpotent parabolic Springer fibres and nilpotent Lusztig varieties. This points the way to a generalization of modified Macdonald polynomials and integral form Macdonald polynomials to all Lie types. But these generalizations are not polynomials, they are elements of the Iwahori-Hecke algebra of the finite Weyl group. This work concerns the generalization of, and connection between, a 1997 paper of Halverson-Ram (which counts points of nilpotent Lusztig varieties over a finite field) and a 2017 paper of Mellit (which counts points of nilpotent parabolic affine Springer fibres over a finite field).

Chris Godsil: Algebraic graph theory and quantum computing

8th January, 2024

This is a 32-lecture course, with each lecture being about 45 minutes, given by Chris Godsil. Note that the 17th lecture was not recorded, but slides are at least available for it. The other 31 lectures are still of interest, but this needs to be known.

This course will provide an introduction to problems in quantum computing that can be studied using tools from algebraic graph theory. The quantum topics will relate to quantum walks and to quantum homomorphisms, automorphisms and colouring. The tools from algebraic graph theory include graphs automorphisms and homomorphisms, spectral decomposition and generating functions.

Prerequisites: I will assume a solid background in linear algebra and knowledge of what a permutation group is. Other topics will be covered in class, or in the notes. I will assume the knowledge of physics I had when I started on this topic, that is, no knowledge.

Alexander Yong: Newell-Littlewood numbers

28th November, 2023

The Newell-Littlewood numbers are defined in terms of the Littlewood-Richardson coefficients from algebraic combinatorics. Both appear in representation theory as tensor product multiplicities for a classical Lie group. This talk concerns the question: Which multiplicities are non-zero? In 1998, Klyachko established common linear inequalities defining both the eigencone for sums of Hermitian matrices and the saturated Littlewood-Richardson cone. We prove some analogues of Klyachko's non-vanishing results for the Newell-Littlewood numbers.

Martha Precup: The cohomology of nilpotent Hessenberg varieties and the dot action representation

10th April, 2023

In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links the combinatorics of chromatic symmetric functions to the geometry of Hessenberg varieties via a permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give a brief overview of that story and discuss how the dot action can be computed in all Lie types using the Betti numbers of certain nilpotent Hessenberg varieties. As an application, we obtain new geometric insight into certain linear relations satisfied by chromatic symmetric functions, known as the modular law.

Greta Panova: Computational complexity in algebraic combinatorics II

8th April, 2023

Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives on their own and brought us some beautiful formulas and combinatorial interpretations.

The flagship hook-length formula counts the number of standard Young tableaux, which also give the dimension of the irreducible Specht modules of the symmetric group. The elegant Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, becoming applicable to Integrable Probability and Statistical Mechanics, and Computational Complexity Theory.

We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, which can formally explain the beauty we see and the difficulties we encounter in finding further formulas and "combinatorial interpretations". In the opposite direction, the 85 year old open problem on Kronecker coefficients of the symmetric group lead to the disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation-theoretic multiplicities in further detail, possibly asymptotically.

Greta Panova: Computational complexity in algebraic combinatorics I

8th April, 2023

Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives on their own and brought us some beautiful formulas and combinatorial interpretations.

The flagship hook-length formula counts the number of standard Young tableaux, which also give the dimension of the irreducible Specht modules of the symmetric group. The elegant Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, becoming applicable to Integrable Probability and Statistical Mechanics, and Computational Complexity Theory.

We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, which can formally explain the beauty we see and the difficulties we encounter in finding further formulas and "combinatorial interpretations". In the opposite direction, the 85 year old open problem on Kronecker coefficients of the symmetric group lead to the disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation-theoretic multiplicities in further detail, possibly asymptotically.