arXiv subjects
Publications and source records attributed to Álvaro Gutiérrez.
We provide character expansions of certain specialisations of multiparameter Hall--Littlewood polynomials of types $\mathrm{B}_n$, $\mathrm{C}_n$, and $\mathrm{BC}_n$ for rectangular shapes. We use these expansions to establish the six Jacobi-Trudi-type identities conjectured by Ole Warnaar in 2025.
We study the class $\#\mathsf{L}$ of functions counting accepting paths of non-deterministic log-space Turing machines and construct methods to prove containment in $\#\mathsf{L}$. We prove that a large number of classical combinatorial and number theoretic functions belong to this class: classical functions from enumerative combinatorics (multinomial coefficients, Catalan numbers, linear extensions of trees, Stirling numbers, etc), algebraic combinatorics (number of standard Young tableaux, etc), discrete geometry, number theoretic functions, representation theoretic multiplicities in a large class of cases. We show that $\mathrm{GL}_2$-plethysm coefficients of bounded length outer partition can be counted by log$^2$-space polytime verifiers. We pose numerous questions and conjectures on $\#\mathsf{L}$ containment and its generalizations, that suggest venues for conditionally disproving $\#\mathsf{P}$-completeness. While studying which combinatorial functions are in $\#\mathsf{P}$ provides a formal way of (dis)proving the existence of combinatorial interpretations, the lower class $\#\mathsf{L}$ serves as an analogue for functions computable in polynomial time.
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.
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.
Populations of autonomous agents deployed in unknown environments (e.g. planetary or deep-sea exploration) must develop shared vocabularies to refer to entities that have no name in any human language. We propose the Neuro-Symbolic Lexical Discovery (NSLD) framework, in which a population of LLM-based agents plays a referential game over out-of-distribution visual referents, autonomously self-organising a shared alien lexicon. Each agent combines a frozen CLIP vision encoder with a private FAISS vector index and a text-only LLM. Crucially, discovered alien words are anchored to natural language via semantic proximity in the embedding space, enlarging the human vocabulary with new perceptually grounded words. Consensus is reached in simulations with populations of up to twenty agents and ten visual referents. Convergence dynamics are characterised through three analytical models achieving R^2 > 0.95, representing a first step towards pre-deployment planning in autonomous exploration missions.
Orchestrating Large Language Models into Multi-Agent Systems (LLM-MAS) has unlocked remarkable reasoning capabilities, yet emergent failures and hallucinations that resist characterisation block their deployment in safety-critical domains -- a gap made legally untenable by emerging AI regulation. Existing evaluation paradigms share a common flaw: centralised judgment creates single points of failure and demands domain-specific expertise. Here we present POIROT, a protocol that repurposes a system's own agents as its diagnostic layer, leveraging the epistemic diversity already present in the architecture. Across evaluated settings, POIROT outperforms single-LLM evaluator baselines, with gains that scale with problem complexity (OR = 1.60, $p = 0.008$), agent count, and fault dimensionality, persisting under compound fault conditions. These results demonstrate that safety oversight need not be externalised: the agents executing a role carry sufficient collective intelligence to audit it. We release POIROT as an open-source library alongside BLAME, a benchmark for fault attribution in safety-critical multi-agent systems.
Plethysm coefficients $\mathsf{a}_{μ[ν]}^λ$ are the structure coefficients of the plethysm of Schur functions $s_μ[s_ν] = \sum_λ \mathsf{a}_{μ[ν]}^λs_λ$. We study a bivariate generating function of plethysm coefficients when $λ$ has bounded length. We show that this generating function is rational. A key step is MacMahon's combinatory analysis. When the bound on the length is $2$ we give an explicit geometric algorithm to compute it using $q$-Ehrhart theory. We give evidence that the generating function is the quantum Ehrhart series of a union of half-open polytopes and show that it satisfies a reciprocity theorem reminiscent of Ehrhart reciprocity. Furthermore, we give a set of linear recursions that completely describe the $\mathrm{SL}_2$-plethysm coefficients.
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 $\mathrm{SL}_2(\mathbb{F})$.
We prove determinantal-Pfaffian formulae that simultaneously generalise the Pfaffian minor summation formula of Ishikawa and Wakayama and Byun's recent minor summation formula. These formulae are based on factorisation formulae for the determinant of the sum of a skew-symmetric matrix and a rank-1 matrix. Applications include a Cauchy-type identity for skew Schur functions.
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 $\mathcal{U}_q(\mathfrak{sl}_2)$.
We say a sequence $f_0, f_1, f_2, \ldots$ of symmetric functions is Schur log-concave if $f_n^2 - f_{n-1}f_{n+1}$ is Schur positive for all $n\ge1$. We conjecture that a very general class of sequences of Schur functions satisfies this property, and show it for sequences of Schur functions indexed by partitions with growing first part and column. Our findings are related to work of Lam, Postnikov and Pylyavskyy on Schur positivity, and of Butler, Sagan, and the second author on $q$-log-concavity.
To find crystals of $\mathfrak{sl}_2$ representations of the form $Λ^n\text{Sym}^r\mathbb{C}^2$ it suffices to solve the combinatorial problem of decomposing Young's lattice into symmetric, saturated chains. We review the literature on this latter problem, and present a strategy to solve it. For $n \le 4$, the strategy recovers recently discovered solutions. We obtain (i) counting formulas for plethystic coefficients, (ii) new recursive formulas for plethysms of Schur functions, and (iii) formulas for the number of constituents of $Λ^n\text{Sym}^r\mathbb{C}^2$.
We show that the combinatorial definitions of King and Sundaram of the symmetric polynomials of types B and C are indeed symmetric, in the sense that they are invariant by the action of the Weyl groups. Our proof is combinatorial and inspired by Bender and Knuth's classic involutions for type A.
We present the first combinatorial proof of the Graham-Pollak Formula for the determinant of the distance matrix of a tree, via sign-reversing involutions and the Lindström-Gessel-Viennot Lemma. Our approach provides a cohesive and unified framework for the understanding of the existing generalizations and $q$-analogues of the Graham-Pollak Formula, and facilitates the derivation of a natural simultaneous generalizations for them.
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.
Let $T = ([n], E)$ be a tree and let $D = ( d(i,j) )_{i, j \le n}$ be the distance matrix of $T$. Let $S\subseteq [n]$. We give the first combinatorial proof for a formula to compute the principal minor of $D$ indexed by $S$, namely $\det D[S]$. This generalizes work of Graham and Pollak, as well as more recent works.
This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions. The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous $ω$ involution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with $s_λ$ preserves flip-symmetry, provided that $λ$ is a partition of two. Explicit formulas for the iterated plethysms $s_2\circ s_b\circ s_a$ and $s_c\circ s_2\circ s_a$, with $a,$ $b,$ and $c$ $\ge$ $2$ allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients.
We give necessary conditions for the positivity of Littlewood-Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if $S^λ(V)$ appears as a summand in the decomposition into irreducibles of $S^μ(S^ν(V))$, then $ν$'s diagram is contained in $λ$'s diagram.