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Mark Wildon

Publications and source records attributed to Mark Wildon.

At least 19 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 $\Delta^{(n,m)}\mathrm{Sym}^d E$ of the affine group scheme of the algebraic group $\mathrm{SL}_2$, where $E$ is the natural representation and $\Delta^{(n,m)}$ 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

Quantum computation and quantum error correction: the theoretical minimum

These notes introduce quantum computation and quantum error correction, emphasising the importance of stabilisers and the mathematical foundations in basic Lie theory. We begin by using the double cover map $\mathrm{SU}_2 \rightarrow \mathrm{SO}_3(\mathbb{R})$ to illustrate the distinction between states and measurements for a single qubit. We then discuss entanglement and CNOT gates, the Deutsch--Jozsa Problem, and finally quantum error correction, using the Steane $[[7,1,3]]$-code as the main example. The necessary background physics of unitary evolution and Born rule measurements is developed as needed. The circuit model is used throughout.

quant-ph↗
arXiv2025

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

Let $\Delta^\lambda$ be the Weyl functor for the partition $\lambda$ 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 $\Delta^{(M,1^N)}\mathrm{Sym}^d E$ 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 Mart\'inez--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--Saalsch\"utz identity with applications to quantum groups

We present a combinatorial proof of the $q$-Pfaff--Saalsch\"utz 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↗
arXiv2025

Two stability theorems on plethysms of Schur functions

The plethysm product of Schur functions corresponds to composing polynomial representations of infinite general linear groups. Finding the plethysm coefficients $\langle s_\nu \circ s_\mu, s_\lambda\rangle$ that express an arbitrary plethysm $s_\nu \circ s_\mu$ as a sum $\sum_\lambda \langle s_\nu \circ s_\mu, s_\lambda \rangle s_\lambda$ of Schur functions is a fundamental open problem in algebraic combinatorics. We prove two stability theorems for plethysm coefficients under the operations of adding and/or joining an arbitrary partition to either $\mu$ or $\nu$. In both theorems $\mu$ may be replaced with an arbitrary skew partition. As special cases we obtain all stability results on the plethysm product of two Schur functions in the literature to date. The proofs are entirely combinatorial using plethystic semistandard tableaux with positive and negative entries.

math.CO↗
arXiv2024

Weak lumping of left-invariant random walks on left cosets of finite groups

Let $G$ be a finite group and let $H$ be a subgroup of $G$. The left-invariant random walk driven by a probability measure $w$ on $G$ is the Markov chain in which from any state $x \in G$, the probability of stepping to $xg \in G$ is $w(g)$. The initial state is chosen randomly according to a given distribution. The walk is said to lump weakly on left cosets if the induced process on $G/H$ is a time-homogeneous Markov chain. We characterise all the initial distributions and weights $w$ such that the walk is irreducible and lumps weakly on left cosets, and determine all the possible transition matrices of the induced Markov chain. In the case where $H$ is abelian we refine our main results to give a necessary and sufficient condition for weak lumping by an explicit system of linear equations on $w$, organized by the double cosets $HxH$. As an application we consider shuffles of a deck of $n$ cards such that repeated observations of the top card form a Markov chain. Such shuffles include the random-to-top shuffle, and also, when the deck is started in a uniform random order, the top-to-random shuffle. We give a further family of examples in which our full theory of weak lumping is needed to verify that the top card sequence is Markov.

math.PR↗
arXiv2024

FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI

We introduce FrontierMath, a benchmark of hundreds of original, exceptionally challenging mathematics problems crafted and vetted by expert mathematicians. The questions cover most major branches of modern mathematics -- from computationally intensive problems in number theory and real analysis to abstract questions in algebraic geometry and category theory. Solving a typical problem requires multiple hours of effort from a researcher in the relevant branch of mathematics, and for the upper end questions, multiple days. FrontierMath uses new, unpublished problems and automated verification to reliably evaluate models while minimizing risk of data contamination. Current state-of-the-art AI models solve under 2% of problems, revealing a vast gap between AI capabilities and the prowess of the mathematical community. As AI systems advance toward expert-level mathematical abilities, FrontierMath offers a rigorous testbed that quantifies their progress.

cs.AI↗
arXiv2024

A new modular plethystic $\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong \Delta^{(2,1^{N-1})} \mathrm{Sym}^d E$

Let $\mathbb{F}$ be a field and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{F})$. Given a vector space $V$, let $\Delta^{(2,1^{N-1})}V$ be the kernel of the multiplication map $\bigwedge^N V \otimes V \rightarrow \bigwedge^{N+1}V$. We construct an explicit $\mathrm{SL}_2(\mathbb{F})$-isomorphism $\mathrm{Sym}^{N-1}E \otimes \bigwedge^{N+1} \mathrm{Sym}^{d+1}E \cong \Delta^{(2,1^{N-1})} \mathrm{Sym}^d E$. This $\mathrm{SL}_2(\mathbb{F})$-isomorphism is a modular lift of the $q$-binomial identity $q^{\frac{N(N-1)}{2}}[N]_q \binom{d+1}{N+1}_q = s_{(2,1^{N-1})}(1,q,\ldots, q^d)$, where $s_{(2,1^{N-1})}$ is the Schur function for the partition $(2,1^{N-1})$. This identity, which follows from our main theorem, implies the existence of an isomorphism when $\mathbb{F}$ is the field of complex numbers but it is notable, and not typical of the general case, that there is an explicit isomorphism defined in a uniform way for any field.

math.RT↗
arXiv2023

The partition algebra and the plethysm coefficients II: ramified plethysm

The plethysm coefficient $p(\nu, \mu, \lambda)$ is the multiplicity of the Schur function $s_\lambda$ in the plethysm product $s_\nu \circ s_\mu$. In this paper we use Schur--Weyl duality between wreath products of symmetric groups and the ramified partition algebra to interpret an arbitrary plethysm coefficient as the multiplicity of an appropriate composition factor in the restriction of a module for the ramified partition algebra to the partition algebra. This result implies new stability phenomenon for plethysm coefficients when the first parts of $\nu$, $\mu$ and $\lambda$ are all large. In particular, it gives the first positive formula in the case when $\nu$ and $\lambda$ are arbitrary and $\mu$ has one part. Corollaries include new explicit positive formulae and combinatorial interpretations for the plethysm coefficients $p((n-b,b), (m), (mn-r,r))$, and $p((n-b,1^b), (m), (mn-r,r))$ when $m$ and $n$ are large.

math.RT↗
arXiv2023

A necessary and sufficient condition for double coset lumping of Markov chains on groups with an application to the random to top shuffle

Let $Q$ be a probability measure on a finite group $G$, and let $H$ be a subgroup of $G$. We show that a necessary and sufficient condition for the random walk driven by $Q$ on $G$ to induce a Markov chain on the double coset space $H\backslash G/H$, is that $Q(gH)$ is constant as $g$ ranges over any double coset of $H$ in $G$. We obtain this result as a corollary of a more general theorem on the double cosets $H \backslash G / K$ for $K$ an arbitrary subgroup of $G$. As an application we study a variation on the $r$-top to random shuffle which we show induces an irreducible, recurrent, reversible and ergodic Markov chain on the relevant double cosets. The transition matrix of the induced walk has remarkable spectral properties: we find its invariant distribution and its eigenvalues and hence determine its rate of convergence.

math.GR↗
arXiv2021

Modular plethystic isomorphisms for two-dimensional linear groups

Let $E$ be the natural representation of the special linear group $\mathrm{SL}_2(K)$ over an arbitrary field $K$. We use the two dual constructions of the symmetric power when $K$ has prime characteristic to construct an explicit isomorphism $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \mathrm{Sym}_\ell \mathrm{Sym}^m E$. This generalises Hermite reciprocity to arbitrary fields. We prove a similar explicit generalisation of the classical Wronskian isomorphism, namely $\mathrm{Sym}_m \mathrm{Sym}^\ell E \cong \bigwedge^m \mathrm{Sym}^{\ell+m-1} E$. We also generalise a result first proved by King, by showing that if $\nabla^\lambda$ is the Schur functor for the partition $\lambda$ and $\lambda^\circ$ is the complement of $\lambda$ in a rectangle with $\ell+1$ rows, then $\nabla^\lambda \mathrm{Sym}^\ell E \cong \nabla^{\lambda^\circ} \mathrm{Sym}_\ell E$. To illustrate that the existence of such `plethystic isomorphisms' is far from obvious, we end by proving that the generalisation $\nabla^\lambda \mathrm{Sym}^\ell E \cong \nabla^{\lambda'} \mathrm{Sym}^{\ell + \ell(\lambda') - \ell(\lambda)}E$ of the Wronskian isomorphism, known to hold for a large class of partitions over the complex field, does not generalise to fields of prime characteristic, even after considering all possible dualities.

math.RT↗
arXiv2021

Involutive random walks on total orders and the anti-diagonal eigenvalue property

This paper studies a family of random walks defined on the finite ordinals using their order reversing involutions. Starting at $x \in \{0,1,\ldots,n-1\}$, an element $y \le x$ is chosen according to a prescribed probability distribution, and the walk then steps to $n-1-y$. We show that under very mild assumptions these walks are irreducible, recurrent and ergodic. We then find the invariant distributions, eigenvalues and eigenvectors of a distinguished subfamily of walks whose transition matrices have the global anti-diagonal eigenvalue property studied in earlier work by Ochiai, Sasada, Shirai and Tsuboi. We prove that this subfamily of walks is characterised by their reversibility. As a corollary, we obtain the invariant distributions and rate of convergence of the random walk on the set of subsets of $\{1,\ldots, m\}$ in which steps are taken alternately to subsets and supersets, each chosen equiprobably. We then consider analogously defined random walks on the real interval $[0,1]$ and use techniques from the theory of self adjoint compact operators on Hilbert spaces to prove analogues of the main results in the discrete case.

math.PR↗
arXiv2019

Plethysms of symmetric functions and representations of $\mathrm{SL}_2(\mathbb{C})$

Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^\lambda \!\mathrm{Sym}^\ell \!E \cong \nabla^\mu \!\mathrm{Sym}^m \! E$ of representations of $\mathrm{SL}_2(\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $\lambda$ and $\mu$ are conjugate partitions and when one of $\lambda$ or $\mu$ is a rectangle. We give a complete classification when $\lambda$ and $\mu$ each have at most two rows or columns or is a hook partition and a partial classification when $\ell = m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when $\nabla^\lambda \!\mathrm{Sym}^\ell \!E$ is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon's enumeration of plane partitions, and prove a new $q$-binomial identity in this setting.

math.RT↗
arXiv2019

A corollary of Stanley's Hook Content Formula

We use the hook lengths of a partition to define two rectangular tableaux. We prove these tableaux have equal multisets of entries, first by elementary combinatorial arguments, and then using Stanley's Hook Content Formula and symmetric polynomials.

math.CO↗
arXiv2018

Plethysms of symmetric functions and highest weight representations

Let $s_\nu \circ s_\mu$ denote the plethystic product of the Schur functions $s_\nu$ and $s_\mu$. In this article we define an explicit polynomial representation corresponding to $s_\nu \circ s_\mu$ with basis indexed by certain `plethystic' semistandard tableaux. Using these representations we prove generalizations of four results on plethysms due to Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give a sufficient condition for the multiplicity $\langle s_\nu \circ s_\mu, s_\lambda\rangle$ to be stable under insertion of new parts into $\mu$ and $\lambda$. We also characterize all maximal and minimal partitions $\lambda$ in the dominance order such that $s_\lambda$ appears in $s_\nu \circ s_\mu$ and determine the corresponding multiplicities using plethystic semistandard tableaux.

math.RT↗
arXiv2018

A proof of the Murnaghan--Nakayama rule using Specht modules and tableau combinatorics

The Murnaghan--Nakayama rule is a combinatorial rule for the character values of symmetric groups. We give a new combinatorial proof by explicitly finding the trace of the representing matrices in the standard basis of Specht modules. This gives an essentially bijective proof of the rule. A key lemma is an extension of a straightening result proved by the second author to skew-tableaux. Our module theoretic methods also give short proofs of Pieri's rule and Young's rule.

math.RT↗
arXiv2018

The multistep homology of the simplex and representations of symmetric groups

The symmetric group on a set acts transitively on its subsets of a given size. We define homomorphisms between the corresponding permutation modules, defined over a field of characteristic two, which generalize the boundary maps from simplicial homology. The main results determine when these chain complexes are exact and when they are split exact. As a corollary we obtain a new explicit construction of the basic spin modules for the symmetric group.

math.RT↗