Seminars in 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).

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.

Pavel Turek: On stable modular plethysms of the natural module of SL2(đť”˝p) in characteristic p

24th January, 2023

To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field k of non-zero characteristic. However, by describing the representation ring of kSL2(𝔽p) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of kSL2(𝔽p) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural kSL2(𝔽p)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are 'close' to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.

Rowena Paget: Plethysm and the Partition Algebra

10th January, 2023

The symmetric group Smn acts naturally on the collection of set partitions of a set of size mn into n sets each of size m. The irreducible constituents of the associated ordinary character are largely unknown; in particular, they are the subject of the longstanding Foulkes Conjecture. There are equivalent reformulations using polynomial representations of infinite general linear groups or using plethysms of symmetric functions. I will review plethysm from these three perspectives before presenting a new approach to studying plethysm: using the Schur-Weyl duality between the symmetric group and the partition algebra. This method allows us to study stability properties of certain plethysm coefficients. This is joint work with Chris Bowman. If time permits, I will also discuss some new results with Chris Bowman and Mark Wildon.

Nicolle González: Higher Rank Rational (q,t)-Catalan Polynomials and a Finite Shuffle Theorem

29th November, 2022

The classical shuffle theorem states that the Frobenius character of the space of diagonal harmonics is given by a certain combinatorial sum indexed by parking functions on square lattice paths. The rational shuffle theorem, conjectured by Gorsky-Negut and proven by Mellit, states that the geometric action on symmetric functions (described by Schiffmmann-Vasserot) of certain elliptic Hall algebra elements P(m,n) yield the bigraded Frobenius character of a certain Sn representation. This character is known as the Hikita polynomial. In this talk I will introduce the higher-rank rational (q,t)-Catalan polynomials and show these are equal to finite truncations of the Hikita polynomial. By generalizing results of Gorsky-Mazin-Vazirani and constructing an explicit bijection between rational semistandard parking functions and affine compositions, I will derive a finite analogue of the rational shuffle theorem in the context of spherical double affine Hecke algebras.

Daniel Orr: Difference Operators for Wreath Macdonald Polynomials

17th October, 2021

very concrete auxiliary algebraic structures that were constructed in order to define them. Later, when Haiman's proof of the Macdonald positivity conjecture revolutionized the subject, the scope of Macdonald theory widened to include the geometry of Hilbert schemes of points in the plane. (For this reason, one should associate ordinary Macdonald polynomials with the Jordan quiver.)

A cyclic quiver generalization of Macdonald polynomials was born in reverse, starting with a geometric conjecture which was made by Haiman and later proved by Bezrukavnikov and Finkelberg. Thus the resulting polynomials, which are known as wreath Macdonald polynomials, arise from the geometry of cyclic quiver Nakajima varieties. Their existence relies on an elusive object known as the Procesi bundle, which is available only by deep and indirect means.

Only recently has direct understanding of wreath Macdonald polynomials begun to emerge, through methods based on the quantum toroidal algebra. In this talk, I will review the origins of (wreath) Macdonald theory and discuss new explicit results on wreath Macdonald polynomials, and anticipated applications, from joint work in progress with Mark Shimozono and Joshua Wen.

Mark Wildon: Plethysms, polynomial representations of linear groups and Hermite reciprocity over an arbitrary field

27th April, 2021

Let E be a 2-dimensional vector space. Over the complex numbers the irreducible polynomial representations of the special linear group SL(E) are the symmetric powers SymrE. Composing polynomial representations, for example to form Sym4 Sym2E, corresponds to the plethysm product on symmetric functions. Expressing such a plethysm as a linear combination of Schur functions has been identified by Richard Stanley as one of the fundamental open problems in algebraic combinatorics. In my talk I will use symmetric functions to prove some classical isomorphisms, such as Hermite reciprocity Symm SymrE ≅ SymrSymmE, and some others discovered only recently in joint work with Rowena Paget. I will then give an overview of new results showing that, provided suitable dualities are introduced, Hermite reciprocity holds over arbitrary fields; certain other isomorphisms (we can prove) have no modular generalization.

Evgeny Mukhin: Supersymmetric analogues of partitions and plane partitions

8th April, 2021

We will explain combinatorics of various partitions arising in the representation theory of quantum toroidal algebras associated to Lie superalgebra 𝔤𝔩(m|n). Apart from being interesting in its own right, this combinatorics is expected to be related to crystal bases, fixed points of the moduli spaces of BPS states, equivariant K-theory of moduli spaces of maps, and other things.

Jieru Zhu: Transitioning between the polytabloid and web bases for the Specht modules

8th March, 2021

SLn webs first emerged in invariant theory and have a recent reformulation by Cautis-Kamnitzer-Morrison (2014). A collection of these webs form a basis of the Specht modules for the symmetric groups. On the other hand, classical construction of the Specht modules uses the polytabloids basis parameterized by standard Young tableaux. Russell-Tymoczko (2020) showed that the transitioning matrix from the polytabloid basis to the web basis is unitriangular. We further proved their conjecture that the upper-triangular entries are positive. The talk will be mostly focused on SL2 webs, with some preliminary results on SL3 webs.

Reuven Hodges: Coxeter combinatorics and spherical Schubert geometry

16th November, 2020

This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety G/B that are spherical for the action of a Levi subgroup. We will see that this conjecture extends and unifies previous sphericality results for Schubert varieties in G/B due to P. Karuppuchamy, J. Stembridge, P. Magyar–J. Weyman-A. Zelevinsky. In type A, the combinatorics of Demazure modules and their key polynomials, multiplicity freeness, and split-symmetry in algebraic combinatorics are employed to prove this conjecture for several classes of Schubert varieties.

Nicolle Gonzalez: A skein-theoretic Carlsson-Mellit algebra

2nd November, 2020

The shuffle conjecture was a big open problem in algebraic combinatorics which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This conjecture was finally solved after 14 years by Carlsson and Mellit by the introduction of a new interesting algebra denoted Aq,t. This algebra arises as an extension of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. In later work by Carlsson, Mellit, and Gorsky this algebra and its representation was realized using parabolic flag Hilbert schemes and was also shown to contain the generators of the elliptic Hall algebra. I will discuss a new topological formulation of Aq,t and its representation over a thickened annulus and a categorification thereof over the derived trace of the Soergel category. This is joint work with Matt Hogancamp.