arXiv subjects

Michał Szwej

Publications and source records attributed to Michał Szwej.

4 recordsLinked to original sources

arXiv2026

A field-independent filtration of plethystic modules for $\mathrm{SL}_2(\mathbb{F})$ that categorifies a product rule for the Cartan subalgebra of $\mathcal{U}_q(\mathfrak{sl}_2)$

We lift a product rule in the Cartan subalgebra of quantum $\mathfrak{sl}_2$ to a filtration of the plethystic representation $Δ^{(n,m)}\mathrm{Sym}^d E$ of the affine group scheme of the algebraic group $\mathrm{SL}_2$, where $E$ is the natural representation and $Δ^{(n,m)}$ the Weyl functor. This is a significant step towards a categorification of quantum $\mathfrak{sl}_2$. Our filtration is an addition to a growing family of field-independent isomorphisms of $\mathrm{SL}_2$ representations that include Hermite reciprocity and the Wronskian isomorphism. It is the first such field-independent result requiring multiple filtration layers. It is proved by combinatorial techniques using the authors' symmetric functions model for Weyl modules.

math.RT↗

arXiv2026

On the generalised Foulkes conjecture for $\mathrm{SL}_2(\mathbb{C})$ under divisibility conditions

The generalised Foulkes conjecture for $\mathrm{SL}_2(\mathbb{C})$ was recently proved under mild divisibility conditions by Raicu, Sam, Weyman, and Yang, who showed surjectivity of the Foulkes--Howe map through a geometric and homological approach. As an immediate corollary, we note partial proofs of related conjectures by Bergeron, Zanello, and Troyka. The main result of this paper is that the dual of the (geometric) Foulkes--Howe map is the (combinatorial) $k$-fold plethystic substitution, which admits an explicit and straightforward definition. We derive several formulas and combinatorial interpretations for its structure constants. We briefly remark on an unexpected corollary that settles a conjecture in condensed matter physics. Finally, we use the combinatorial properties of the $k$-fold map to propose candidate maps towards other variants of Foulkes' conjecture.

math.CO↗

arXiv2026

Modular isomorphisms of $\mathrm{SL}_2(\mathbb{F})$-plethysms for Weyl modules labelled by hook partitions

Let $Δ^λ$ be the Weyl functor for the partition $λ$ and let $E$ be the natural $2$-dimensional representation of $\mathrm{SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. We give an explicit isomorphism showing that any $\mathrm{SL}_2(\mathbb{F})$-plethysm $Δ^{(M,1^N)}\mathrm{Sym}^d E$ factors as a tensor product of two simpler $\mathrm{SL}_2(\mathbb{F})$-plethysms, each defined using only symmetric powers. This result categorifies Stanley's Hook Content Formula for hook-shaped partitions and proves a conjecture of Martínez--Wildon (2024). In a similar spirit we categorify the classical binomial identity $\binom{a}{b}\binom{b}{c}=\binom{a}{c}\binom{a-c}{b-c}$, obtaining a new family of $\mathrm{SL}_2(\mathbb{F})$-isomorphisms between tensor products of plethysms. Our methods are characteristic independent and provide a framework that is broadly applicable to the study of isomorphisms between plethystic representations of $\mathrm{SL}_2(\mathbb{F})$.

math.RT↗

arXiv2025

A new bijective proof of the $q$-Pfaff--Saalschütz identity with applications to quantum groups

We present a combinatorial proof of the $q$-Pfaff--Saalschütz identity by a composition of explicit bijections, in which $q$-binomial coefficients are interpreted as counting subspaces of $\mathbb{F}_q$-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig's integral form $\mathcal{U}_{\mathbb{Z}[q, q^{-1}]}(\mathfrak{sl}_2)$ of the Cartan subalgebra of the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$.

math.CO↗