Searcharxiv⌕ Search

arXiv subjects

Jonathan Brundan

Publications and source records attributed to Jonathan Brundan.

At least 19 recordsLinked to original sources
arXiv2025

Yangians and degenerate affine Schur algebras

Drinfeld's degenerate affine analog of Schur-Weyl duality relates representations of the degenerate affine Hecke algebra $AH_r$ to representations of the Yangian $Y_n$. One way to understand the construction is to introduce an intermediate algebra $AS(n,r)$, the degenerate affine Schur algebra, which appears both as the endomorphism algebra of an induced tensor space over $AH_r$, and as the image of a homomorphism $D_{n,r}:Y_n \rightarrow AS(n,r)$. In this paper, we describe $D_{n,r}$ using a diagrammatic calculus. Then we use a theorem of Drinfeld to compute $\ker D_{n,r}$ when $n > r$, thereby giving a presentation of $AS(n,r)$ in these cases. We formulate a conjecture in the remaining cases. Finally, we apply results of Arakawa to develop some of the representation theory of $AS(n,r)$.

math.RT↗
arXiv2025

Isomeric Heisenberg and Kac-Moody categorification I

We develop a general framework for studying Abelian categories arising in isomeric representation theory, that is, representation theory broadly related to the supergroup Q(n). In this first part, we introduce notions of isomeric Heisenberg categorification and isomeric Kac-Moody categorication, and explain how to pass from the former to the latter. This is analogous to the passage from Heisenberg categorification to Kac-Moody categorification developed in our previous work with Webster.

math.RT↗
arXiv2025

An update on Heisenberg and Kac-Moody categorification

Heisenberg categories act on many Abelian categories appearing in type A representation theory. There is also a general procedure to construct from a Heisenberg action another action of a Kac-Moody 2-category for some associated Cartan matrix. One of the adjunctions on the Kac-Moody side is matched up in an easy way with adjunctions on the Heisenberg side, but the second adjunction is much harder to describe. In this paper, we derive explicit formulae for this difficult adjunction, leading to some further simplifications to the existing theory.

math.RT↗
arXiv2025

Categorification of quasi-split iquantum groups

We introduce a new family of graded 2-categories generalizing the 2-quantum groups introduced by Khovanov, Lauda and Rouquier. We use them to categorify quasi-split iquantum groups in all symmetric types.

math.QA↗
arXiv2024

Cyclotomic nil-Brauer and Singular Soergel bimodules of type D

We introduce a new family of monoidal categories which are cyclotomic quotients of the nil-Brauer category. We construct a monoidal functor from the cyclotomic nil-Brauer category to another monoidal category constructed from singular Soergel bimodules of type D. We conjecture that our functor is an equivalence of categories. Although we can prove neither fullness nor faithfulness at this point, we are able to show that the functor induces an isomorphism at the level of Grothendieck rings. We compute these rings and their canonical bases, and give diagrammatic descriptions of the corresponding primitive idempotents.

math.RT↗
arXiv2024

The $q$-Schur category and polynomial tilting modules for quantum $GL_n$

The $q$-Schur category is a $\mathbb{Z}[q,q^{-1}]$-linear monoidal category closely related to the $q$-Schur algebra. We explain how to construct it from coordinate algebras of quantum $GL_n$ for all $n \geq 0$. Then we use Donkin's work on Ringel duality for $q$-Schur algebras to make precise the relationship between the $q$-Schur category and an integral form for the $U_q\mathfrak{gl}_n$-web category of Cautis, Kamnitzer and Morrison. We construct explicit integral bases for morphism spaces in the latter category, and extend the Cautis-Kamnitzer-Morrison theorem to polynomial representations of quantum $GL_n$ at a root of unity over a field of any characteristic.

math.QA↗
arXiv2023

Nil-Brauer categorifies the split iquantum group of rank one

We prove that the Grothendieck ring of the monoidal category of finitely generated graded projective modules for the nil-Brauer category is isomorphic to an integral form of the split iquantum group of rank one. Under this isomorphism, the indecomposable graded projective modules correspond to the icanonical basis. We also derive character formulae for irreducible graded modules and deduce various branching rules.

math.QA↗
arXiv2023

Graded triangular bases

This article develops a practical technique for studying representations of $\Bbbk$-linear categories arising in the categorification of quantum groups. We work in terms of locally unital algebras which are $\mathbb{Z}$-graded with graded pieces that are finite-dimensional and bounded below, developing a theory of graded triangular bases for such algebras. The definition is a graded extension of the notion of triangular basis introduced in previous joint work with Stroppel. However, in the general graded setting, finitely generated projective modules often fail to be Noetherian, so that existing results from the study of highest weight categories are not directly applicable. Nevertheless, we show that there is still a good theory of standard modules. In motivating examples arising from Kac-Moody 2-categories, these modules categorify the PBW bases for the modified forms of quantum groups constructed by Wang.

math.RT↗
arXiv2023

The nil-Brauer category

We introduce the nil-Brauer category and prove a basis theorem for its morphism spaces. This basis theorem is an essential ingredient required to prove that nil-Brauer categorifies the split iquantum group of rank one. As this iquantum group is a basic building block for $\imath$-quantum groups of higher rank, we expect that the nil-Brauer category will play a role in future developments related to the categorification of quantum symmetric pairs.

math.RT↗
arXiv2022

Odd Grassmannian bimodules and derived equivalences for spin symmetric groups

We prove odd analogs of results of Chuang and Rouquier on sl(2)-categorification. Combined also with recent work of the second author with Livesey, this allows us to complete the proof of Brou\'e's Abelian Defect Conjecture for the double covers of symmetric groups. The article also develops the theory of odd symmetric functions initiated a decade ago by Ellis, Khovanov and Lauda. A key role in our approach is played by a 2-category consisting of odd Grassmannian bimodules over superalgebras which are odd analogs of equivariant cohomology algebras of Grassmannians. This is the odd analog of the category of Grassmannian bimodules which was at the heart of Lauda's independent approach to categorification of sl(2). We also construct an action of the odd Kac-Moody 2-category of sl(2) on the 2-category of odd Grassmannian bimodules, and use this to give a new proof of its non-degeneracy.

math.RT↗
arXiv2021

A new approach to the representation theory of the partition category

We explain a new approach to the representation theory of the partition category based on a reformulation of the definition of the Jucys-Murphy elements introduced originally by Halverson and Ram and developed further by Enyang. Our reformulation involves a new graphical monoidal category, the affine partition category, which is defined here as a certain monoidal subcategory of Khovanov's Heisenberg category. We use the Jucys-Murphy elements to construct some special projective functors, then apply these functors to give self-contained proofs of results of Comes and Ostrik on blocks of Deligne's category Rep(S_t).

math.RT↗
arXiv2020

Quantum Frobenius Heisenberg categorification

We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring. When $A$ is trivial, i.e. it equals the ground ring, these categories recover the quantum Heisenberg categories introduced in our previous work, and when the central charge $k$ is zero they yield generalizations of the affine HOMFLY-PT skein category. By exploiting some natural categorical actions of $\mathcal{H}\textit{eis}_k(A;z,t)$ on generalized cyclotomic quotients, we prove a basis theorem for morphism spaces.

math.RT↗
arXiv2020

Foundations of Frobenius Heisenberg categories

We describe bases for the morphism spaces of the Frobenius Heisenberg categories associated to a symmetric graded Frobenius algebra, proving several open conjectures. Our proof uses a categorical comultiplication and generalized cyclotomic quotients of the category. We use our basis theorem to prove that the Grothendieck ring of the Karoubi envelope of the Frobenius Heisenberg category recovers the lattice Heisenberg algebra associated to the Frobenius algebra.

math.RT↗
arXiv2020

Heisenberg and Kac-Moody categorification

We show that any Abelian module category over the (degenerate or quantum) Heisenberg category satisfying suitable finiteness conditions may be viewed as a 2-representation over a corresponding Kac-Moody 2-category (and vice versa). This gives a way to construct Kac-Moody actions in many representation-theoretic examples which is independent of Rouquier's original approach via `control by K_0.' As an application, we prove an isomorphism theorem for generalized cyclotomic quotients of these categories, extending the known isomorphism between cyclotomic quotients of type A affine Hecke algebras and quiver Hecke algebras.

math.RT↗
arXiv2019

Whittaker coinvariants for $\mathrm{GL}(m|n)$

Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. In this paper we study the {\em Whittaker coinvariants functor}, which is an exact functor from category $\mathcal O$ for $\mathfrak{gl}_{m|n}(\mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergel's functor $\mathbb V$ in the setting of category $\mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $\mathcal O$ up to Morita/derived equivalence.

math.RT↗
arXiv2018

On the definition of quantum Heisenberg category

We introduce a diagrammatic monoidal category $\mathcal{H}eis_k(z,t)$ which we call the quantum Heisenberg category, here, $k \in \mathbb{Z}$ is "central charge" and $z$ and $t$ are invertible parameters. Special cases were known before: for central charge $k=-1$ and parameters $z = q-q^{-1}$ and $t = -z^{-1}$ our quantum Heisenberg category may be obtained from the deformed version of Khovanov's Heisenberg category introduced by Licata and the second author by inverting its polynomial generator, while $\mathcal{H}eis_0(z,t)$ is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in $\mathcal{H}eis_k(z,t)$.

math.RT↗
arXiv2018

The degenerate Heisenberg category and its Grothendieck ring

The degenerate Heisenberg category $\mathcal{H}eis_k$ is a strict monoidal category which was originally introduced in the special case $k=-1$ by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi envelope of his category is isomorphic to a certain $\mathbb{Z}$-form for the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra specialized at central charge $-1$. We prove this conjecture and extend it to arbitrary central charge $k \in \mathbb{Z}$. We also explain how to categorify the comultiplication (generically).

math.CT↗