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Kay Jin Lim

Publications and source records attributed to Kay Jin Lim.

At least 19 recordsLinked to original sources
arXiv2026

Towards fault-tolerance with universal phase-error-transparent gates for high-spin cat codes

High-dimensional nuclear spins offer a hardware-efficient route to quantum error correction (QEC), with the spin cat code providing intrinsic robustness against phase errors -- the dominant noise channel in donor-in-silicon architectures. However, realizing the full potential of this encoding requires gate operations that preserve its error-correcting properties. In this work, we construct a universal logical gate set that is error-transparent (ET) to phase errors, and discuss its practical implementations and challenges. The ET gates ensure that phase errors occurring stochastically during gate operations are propagated in a systematically traceable manner and remain correctable in a subsequent QEC step. Among the universal gate set constructed, we identify the logical $X$ gate as the primary challenge and discuss potential realization schemes. In addition, to fully leverage the spin cat code's advantage over an unencoded qubit, multi-tone microwave driving of the logical $CZ$ gate is essential. Our simulations show that ET gates significantly outperform non-ET gates and may be necessary to surpass the break-even point. We further show how logical measurement and recovery can be constructed from ET operations, and explain why state-preparation cannot be made ET. In particular, ET measurement in the computational basis is realizable via spin parity measurement, and that error correction circuits constructed from ET operations achieve optimal error correction capacity. Our work charts a concrete path toward full fault-tolerant quantum computation with high-dimensional nuclear spin systems.

quant-ph↗
arXiv2026

Simple modules for affine nilCoxeter algebras

We study the representation theory of the affine nilCoxeter algebra $A$ of type $\tilde A_{n-1}$, over a field $k$ of any characteristic. Our main theorem states that this is a Noetherian prime affine PI algebra of PI degree $n!$. As a consequence, the simple $A$-modules are all finite dimensional, and the maximum dimension of a simple module is $n!$ over a suitable finite extension of $k$. To achieve this, we investigate a large commutative subalgebra $C$ which is finitely generated as an algebra and over which $A$ is finitely generated as a module. We show that the associated primes of $C$ are minimal primes, and there are $n!$ of them, regularly permuted by $\mathfrak{S}_n$. The algebra $R=C^{\mathfrak{S}_n}$ is equal to the centre of $A$, and isomorphic to $C/\mathfrak{p}$ for each of the minimal primes $\mathfrak{p}$. We prove that the ring $R$ is isomorphic to $k[X_1,\dots,X_{n-1}]^{\mu_n}$, where $\mu_n$ is the finite group scheme of $n$th roots of unity, acting so that $X_i$ has degree $i$ modulo $n$. The ring $R$ is Cohen--Macaulay, and is Gorenstein if and only if $n$ is odd or $n=2$. It is a toric ring, with divisor class group $\mathsf{Cl}(R)\cong\mathbb{Z}/n$, and every projective $R$-module is free.

math.RT↗
arXiv2026

Tensor Powers of Indecomposable Modules for $F\mathfrak{S}_p$

In the earlier paper arXiv:2603.11533, the authors gave an explicit tensor product formula, modulo projectives, for modules over the group algebra $F\mathfrak{S}_p$. In this paper, we use this formula to study tensor powers of such modules. In certain cases, we give combinatorial descriptions of the multiplicities of the indecomposable summands appearing in their direct-sum decompositions. We also study the asymptotic behaviour of these multiplicities as the underlying prime tends to infinity.

math.RT↗
arXiv2026

The Representation Type of the Descent Algebras

Schocker classified the representation type of the descent algebra of type $\mathbb{A}$ over any field of characteristic zero. In an earlier paper, the authors extended this classification for type $\mathbb{A}$ to fields of positive characteristic. In this paper, we complete the classification for all other types except for $\mathbb{E}_8$. The proof for type $\mathbb{B}$ is entirely theoretical, while some small cases in type $\mathbb{D}$ and the exceptional types require computer computation to determine their Ext-quivers.

math.RT↗
arXiv2026

Tensor Product and the Stable Green Ring of the Symmetric Group Algebra $F\mathfrak{S}_p$

We give an explicit formula for the decomposition of the tensor product of any two indecomposable non-projective modules for the symmetric group algebra $F \mathfrak{S}_p$ modulo projective modules. In particular, we show that the tensor product of two simple modules is semisimple modulo projectives. We also compute the Benson--Symonds invariants for all such indecomposable non-projective modules.

math.RT↗
arXiv2024

Small modules with interesting rank varieties

This paper focuses on the rank varieties for modules over a group algebra $\mathbb{F}E$ where $E$ is an elementary abelian $p$-group and $p$ is the characteristic of an algebraically closed field $\mathbb{F}$. In the first part, we give a sufficient condition for a Green vertex of an indecomposable module containing an elementary abelian $p$-group $E$ in terms of the rank variety of the module restricted to $E$. In the second part, given a homogeneous algebraic variety $V$ , we explore the problem on finding a small module with rank variety $V$ . In particular, we examine the simple module $D^{(kp-p+1,1^{p-1})}$ for the symmetric group $\mathfrak{S}_{kp}$.

math.RT↗
arXiv2024

The Representation Type of the Descent Algebras of Type $\mathbb{A}$

We classify the representation type of the descent algebras of type $\A$ in the positive characteristic case. The algebras have finite representation type only for a few small degrees; otherwise, they are wild. Our main reduction method relies on a surjective algebra homomorphism from a descent algebra of type $\A$ to another of lower degree. For small degree cases, we employ techniques from the representation theory of finite-dimensional algebras.

math.RT↗
arXiv2024

Projective Modules and Cohomology for Integral Basic Algebras

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. However, for certain algebras, for example the group algebras, they behave the same way as the characteristic zero case at "good enough" prime. In this paper, we initiate the study of this topic by imposing increasingly strong hypotheses on basic algebras. When the algebras satisfy the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. The examples include the Solomon descent algebras of finite Coxeter groups at large enough primes, nilCoxeter algebra, and certain finite semigroup algebras at an arbitrary prime.

math.RT↗
arXiv2023

Modular Idempotents for the Descent Algebras of Type A and Higher Lie Powers and Modules

The article focuses on four aspects related to the descent algebras of type $A$. They are modular idempotents, higher Lie powers, higher Lie modules and the right ideals of the symmetric group algebras generated by the Solomon's descent elements. More precisely, we give a construction for the modular idempotents, describe the dimension and character for higher Lie powers and study the structures of the higher Lie modules and the right ideals both in the ordinary and modular cases.

math.RT↗
arXiv2021

Young's seminormal basis vectors and their denominators

We study Young's seminormal basis vectors of the dual Specht modules of the symmetric group, indexed by a certain class of standard tableaux, and their denominators. These vectors include those whose denominators control the splitting of the canonical morphism $Δ(λ+μ) \to Δ(λ) \otimes Δ(μ)$ over $\mathbb{Z}_{(p)}$, where $Δ(ν)$ is the Weyl module of the classical Schur algebra labelled by $ν$.

math.RT↗
arXiv2020

On signed $p$-Kostka matrices

We show that the signed $p$-Kostka numbers depend just on $p$-Kostka numbers and the multiplicities of projective indecomposable modules in certain signed Young permutation modules. We then examine the signed $p$-Kostka number $k_{(α|β),(λ|pμ)}$ in the case when $|β|=p|μ|$. This allows us to explicitly describe the multiplicities of direct summands of a signed Young permutation module lying in the principal block of $F\mathfrak{S}_{mp}$ in terms of the $p$-Kostka numbers.

math.RT↗
arXiv2020

Jantzen filtration of Weyl modules, product of Young symmetrizers and denominator of Young's seminormal basis

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p>0$, $Δ(λ)$ denote the Weyl module of $G$ of highest weight $λ$ and $ι_{λ,μ}:Δ(λ+μ)\to Δ(λ)\otimesΔ(μ)$ be the canonical $G$-morphism. We study the split condition for $ι_{λ,μ}$ over $\mathbb{Z}_{(p)}$, and apply this as an approach to compare the Jantzen filtrations of the Weyl modules $Δ(λ)$ and $Δ(λ+μ)$. In the case when $G$ is of type $A$, we show that the split condition is closely related to the product of certain Young symmetrizers and, under some mild conditions, is further characterized by the denominator of a certain Young's seminormal basis vector. We obtain explicit formulas for the split condition in some cases.

math.RT↗
arXiv2018

Straightening rule for an $m'$-truncated polynomial ring

We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime $p$ the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pairs of partitions. As a result, for each partition $λ$, we discovered the number of compositions $δ$ such that $δ$ can be rearranged to $λ$ and whose partial sums of $δ$ are not divisible by $p$.

math.RT↗
arXiv2018

Homomorphisms from Specht Modules to Signed Young Permutation Modules

We construct a class $Θ_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^λ$ to a signed permutation module $M_{\mathbb{Z}}(α|β)$ which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any $ϕ\in \operatorname{Hom}_{\mathbb{Z}\mathfrak{S}_{n}}\big(S_{\mathbb{Z}}^λ, M_{\mathbb{Z}}(α|β)\big)$ lies in the $\mathbb{Q}$-span of $Θ_{\text{sstd}}$, a subset of $Θ_{\mathscr{R}}$ corresponding to semistandard $λ$-tableaux of type $(α|β)$. We also study the conditions for which $Θ^{\mathbb{F}}_{\mathrm{sstd}}$ - a subset of $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^λ,M_{\mathbb{F}}(α|β)\big)$ induced by $Θ_{\mathrm{sstd}}$ - is linearly independent, and show that it is a basis for $\operatorname{Hom}_{\mathbb{F}\mathfrak{S}_{n}}\big(S_{\mathbb{F}}^λ,M_{\mathbb{F}}(α|β)\big)$ when $\mathbb{F}\mathfrak{S}_{n}$ is semisimple.

math.RT↗
arXiv2017

On the Brauer constructions and generic Jordan types of Young modules

Let p be a prime number. We study the dimensions of Brauer constructions of Young and Young permutation modules with respect to p-subgroups of the symmetric groups. They depend only on partitions labelling the modules and the orbits of the action of the p-subgroups, and are related to their generic Jordan types. We obtain some reductive formulae and, in the case of two-part partitions, make some explicit calculation.

math.RT↗
arXiv2017

Signed Young Modules and Simple Specht Modules

By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are parametrized by certain pairs of partitions. The main result of this article establishes the signed Young module labels of simple Specht modules. Along the way we prove a number of results concerning indecomposable signed Young modules that are of independent interest. In particular, we determine the label of the indecomposable signed Young module obtained by tensoring a given indecomposable signed Young module with the sign representation. As consequences, we obtain the Green vertices, Green correspondents, cohomological varieties, and complexities of all simple Specht modules and a class of simple modules of symmetric groups, and extend the results of Gill on periodic Young modules to periodic indecomposable signed Young modules.

math.RT↗
arXiv2016

On signed p-Kostka numbers and the indecomposable signed Young permutation modules

We prove the existence and main properties of signed Young modules for the symmetric group, using only basic facts about symmetric group representations and the Brou{é} correspondence. We then prove new reduction theorems for the signed $p$-Kostka numbers, defined to be the multiplicities of signed Young modules as direct summands of signed Young permutation modules. We end by classifying the indecomposable signed Young permutation modules and determining their endomorphism algebras.

math.RT↗