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Jay Taylor

Publications and source records attributed to Jay Taylor.

At least 19 recordsLinked to original sources
arXiv2026

Probabilistic Atomic Swaps for Bitcoin and Friends

Atomic swaps are a fundamental primitive for the trustless exchange of digital assets across blockchains: they guarantee that either both parties receive the agreed assets or neither party transfers. While this all-or-nothing guarantee is powerful, it also imposes an inherent determinism that rules out exchanges whose intended outcome is probabilistic. As a result, existing atomic swaps cannot realize trustless exchanges in which one party pays for a fixed chance of receiving a larger asset or reward, as in lotteries, randomized allocation mechanisms, and probabilistic cross-chain trades. We introduce probabilistic swaps, a new cryptographic primitive that extends atomic swaps to the probabilistic setting. In a probabilistic swap, one party's transfer is executed with a fixed, publicly specified probability embedded in the protocol and cannot be biased by either party. This yields a trustless mechanism for randomized exchange with verifiable odds and no trusted intermediary. Our construction combines adaptor signatures with oblivious pseudorandom functions (OPRFs) to realize the desired probabilistic outcome while ensuring that neither party can predict or bias it in advance. Along the way, we introduce a new mechanism for the atomic exchange of OPRF evaluations for payments, which may be of independent interest. A key feature of our approach is that it preserves the minimal on-chain footprint of modern atomic-swap protocols. The protocol relies only on standard Bitcoin scripts, such as digital signatures and timelocks, and is deployable on any blockchain that already supports atomic swaps. Consequently, probabilistic swaps are indistinguishable from ordinary on-chain transactions, which helps preserve privacy and fungibility. We provide formal security foundations and demonstrate practicality through a probabilistic swap in the Bitcoin testnet and in the Lightning Network.

cs.CR↗
arXiv2025

Towards the inductive McKay--Navarro Condition for groups of Lie type

We gather tools for proving the inductive McKay--Navarro (or Galois--McKay) condition for groups of Lie type and odd primes. We use this to establish a bijection in the case of quasisimple groups of Lie type A satisfying the equivariance properties needed for the condition. We also prove the inductive conditions for the subset of unipotent characters.

math.RT↗
arXiv2024

Representation growth of Fuchsian groups and modular forms

Let $\Gamma$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $\Gamma$. Then $|\mathrm{Hom}(\Gamma,\mathrm{GL}_n(q))|\sim c_{q,n} q^{(1-\chi(\Gamma))n^2}$, where $c_{q,n}$ is periodic in $n$. As a function of $q$, $c_{q,n}$ can be expressed as a Puiseux series in $1/q$ whose coefficients are periodic in $n$ and $q$. Moreover, this series is essentially the $q$-expansion of a meromorphic modular form of half-integral weight.

math.GR↗
arXiv2023

Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne--Michel's unique Jordan decomposition in the connected center case.

math.GR↗
arXiv2022

Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture

For every integer $k$ there exists a bound $B=B(k)$ such that if the characteristic polynomial of $g\in \operatorname{SL}_n(q)$ is the product of $\le k$ pairwise distinct monic irreducible polynomials over $\mathbb{F}_q$, then every element $x$ of $\operatorname{SL}_n(q)$ of support at least $B$ is the product of two conjugates of $g$. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions $(p,q)$, in the special case that $n=p$ is prime, if $g$ has order $\frac{q^p-1}{q-1}$, then every non-scalar element $x \in \operatorname{SL}_p(q)$ is the product of two conjugates of $g$. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

math.RT↗
arXiv2020

Harish-Chandra Cuspidal Pairs

The irreducible characters of a finite reductive group are partitioned into Harish-Chandra series that are labelled by cuspidal pairs. In this note, we describe how one can algorithmically calculate those cuspidal pairs using results of Lusztig.

math.RT↗
arXiv2020

Galois Automorphisms and Classical Groups

In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of Galois automorphisms on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups.

math.RT↗
arXiv2019

Unitriangular Shape of Decomposition Matrices of Unipotent Blocks

We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.

math.RT↗
arXiv2018

Lusztig Induction, Unipotent Supports, and Character Bounds

Recently, a strong exponential character bound has been established in [3] for all elements $g \in \mathbf{G}^F$ of a finite reductive group $\mathbf{G}^F$ which satisfy the condition that the centraliser $C_{\mathbf{G}}(g)$ is contained in a $(\mathbf{G},F)$-split Levi subgroup $\mathbf{M}$ of $\mathbf{G}$ and that $\mathbf{G}$ is defined over a field of good characteristic. In this paper, assuming a weak version of Lusztig's conjecture relating irreducible characters and characteristic functions of character sheaves holds, we considerably generalize this result by removing the condition that $\mathbf{M}$ is split. This assumption is known to hold whenever $Z(\mathbf{G})$ is connected or when $\mathbf{G}$ is a special linear or symplectic group and $\mathbf{G}$ is defined over a sufficiently large finite field.

math.RT↗
arXiv2017

Principal $2$-Blocks and Sylow $2$-Subgroups

Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$. Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal $2$-block of $G$ are fixed by a certain Galois automorphism $\sigma \in \mathrm{Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$. Recent work of Navarro-Vallejo has reduced this conjecture to a problem about finite simple groups. We show that their conjecture holds for all finite simple groups, thus establishing the conjecture for all finite groups.

math.RT↗
arXiv2017

The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups

We investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition we obtain a parameterisation of the isomorphism classes of all root data. By working at the level of root data we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms such embeddings were constructed by Benjamin Martin. In an unpublished manuscript Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. By using our investigations into root data we give new proofs of Asai's results and generalise them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.

math.RT↗
arXiv2017

On The Mackey Formula for Connected Centre Groups

Let $\mathbf{G}$ be a connected reductive algebraic group over $\overline{\mathbb{F}}_p$ and let $F : \mathbf{G} \to \mathbf{G}$ be a Frobenius endomorphism endowing $\mathbf{G}$ with an $\mathbb{F}_q$-rational structure. Bonnaf\'e--Michel have shown that the Mackey formula for Deligne--Lusztig induction and restriction holds for the pair $(\mathbf{G},F)$ except in the case where $q = 2$ and $\mathbf{G}$ has a quasi-simple component of type $\sf{E}_6$, $\sf{E}_7$, or $\sf{E}_8$. Using their techniques we show that if $q = 2$ and $Z(\mathbf{G})$ is connected then the Mackey formula holds unless $\mathbf{G}$ has a quasi-simple component of type $\sf{E}_8$. This establishes the Mackey formula, for instance, in the case where $(\mathbf{G},F)$ is of type $\sf{E}_7(2)$. Using this, together with work of Bonnaf\'e--Michel, we can conclude that the Mackey formula holds on the space of unipotently supported class functions if $Z(\mathbf{G})$ is connected.

math.RT↗
arXiv2017

On Self-Normalising Sylow $2$-Subgroups in Type A

Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalising Sylow $2$-subgroup, which is given in terms of the ordinary irreducible characters of $G$. The first-named author has reduced the proof of this conjecture to showing that certain related statements hold when $G$ is quasisimple. In this article we show that these conditions are satisfied when $G/Z(G)$ is $\mathrm{PSL}_n(q)$, $\mathrm{PSU}_n(q)$, or a simple group of Lie type defined over a finite field of characteristic $2$.

math.RT↗
arXiv2016

Action of Automorphisms on Irreducible Characters of Symplectic Groups

Assume $G$ is a finite symplectic group $\mathrm{Sp}_{2n}(q)$ over a finite field $\mathbb{F}_q$ of odd characteristic. We describe the action of the automorphism group $\mathrm{Aut}(G)$ on the set $\mathrm{Irr}(G)$ of ordinary irreducible characters of $G$. This description relies on the equivariance of Deligne--Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Sp\"ath's criterion for the inductive McKay condition holds for the irreducible characters of $\mathrm{Sp}_{2n}(q)$.

math.RT↗
arXiv2016

A Note on Skew Characters of Symmetric Groups

In previous work Regev used part of the representation theory of Lie superalgebras to compute the values of a character of the symmetric group whose decomposition into irreducible constituents is described by semistandard $(k,\ell)$-tableaux. In this short note we give a new proof of Regev's result using skew characters.

math.RT↗
arXiv2014

Evaluating Characteristic Functions of Character Sheaves at Unipotent Elements

Assume $\mathbf{G}$ is a connected reductive algebraic group defined over an algebraic closure $\mathbb{K} = \overline{\mathbb{F}}_p$ of the finite field of prime order $p>0$. Furthermore, assume that $F : \mathbf{G} \to \mathbf{G}$ is a Frobenius endomorphism of $\mathbf{G}$. In this article we give a formula for the value of any $F$-stable character sheaf of $\mathbf{G}$ at a unipotent element. This formula is expressed in terms of class functions of $\mathbf{G}^F$ which are supported on a single unipotent class of $\mathbf{G}$. In general these functions are not determined, however we give an expression for these functions under the assumption that $Z(\mathbf{G})$ is connected, $\mathbf{G}/Z(\mathbf{G})$ is simple and $p$ is a good prime for $\mathbf{G}$. In this case our formula is completely explicit.

math.RT↗
arXiv2014

Generalised Gelfand-Graev Representations in Small Characteristics

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ of the finite field of prime order $p$ and let $F : \mathbf{G} \to \mathbf{G}$ be a Frobenius endomorphism with $G = \mathbf{G}^F$ the corresponding $\mathbb{F}_q$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig, which decomposes Kawanaka's Generalised Gelfand-Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, Lusztig's results are only valid under the assumption that $p$ is large enough. In this article we show that Lusztig's formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, resp., character sheaf of $\mathbf{G}$, has a unique wave front set, resp., unipotent support, whenever $p$ is good for $\mathbf{G}$.

math.RT↗