ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 20 Jun 2024 12:25:28 GMT2024-06-20T12:25:28Z5091- Young's seminormal basis vectors and their denominatorshttps://scholarbank.nus.edu.sg/handle/10635/226583Title: Young's seminormal basis vectors and their denominators
Authors: Fang, Ming; Lim, Kay Jin; Tan, Kai Meng
Abstract: 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 $\Delta(\lambda+\mu) \to \Delta(\lambda)
\otimes \Delta(\mu)$ over $\mathbb{Z}_{(p)}$, where $\Delta(\nu)$ is the Weyl
module of the classical Schur algebra labelled by $\nu$.
Mon, 01 Nov 2021 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2265832021-11-01T00:00:00Z
- Specht modules with abelian verticeshttps://scholarbank.nus.edu.sg/handle/10635/104179Title: Specht modules with abelian vertices
Authors: Lim, K.J.
Abstract: In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily p 2-cores where p is the characteristic of the underlying field. Furthermore, in the case of p≥3, or p=2 and μ is 2-regular, we show that the complexity of the Specht module S μ is precisely the p-weight of the partition μ. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module S (pp) for p≥3. © 2011 Springer Science+Business Media, LLC.
Wed, 01 Feb 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041792012-02-01T00:00:00Z
- The Schur functor on tensor powershttps://scholarbank.nus.edu.sg/handle/10635/104346Title: The Schur functor on tensor powers
Authors: Lim, K.J.; Tan, K.M.
Abstract: Let M be a left module for the Schur algebra S(n, r), and let s ε ℤ +. Then M ⊗s is a(S(n, rs), FS s)-bimodule, where the symmetric group S sson s letters acts on the right by place permutations. We show that the Schur functor f rs sends M ⊗s to the (FS rs, FS s)-bimodule FS rs⊗ F(SrSs). As a corollary, we obtain the image under the Schur functor of the Lie power L s(M), exterior power s(M)of M and symmetric power S s(M). © 2011 Springer Basel AG.
Wed, 01 Feb 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043462012-02-01T00:00:00Z
- The complexities of some simple modules of the symmetric groupshttps://scholarbank.nus.edu.sg/handle/10635/104268Title: The complexities of some simple modules of the symmetric groups
Authors: Lim, K.J.; Tan, K.M.
Abstract: We determine the complexities of the simple modules lying in two well-known classes of blocks of symmetric group algebras in characteristic p, namely the Rouquier blocks of p-weight w, with w < p, and the defect 2 blocks. For the former, we show that the simple modules have complexity w. For the latter, we show that D(p+1,1p-1) has complexity 1, while the other simple modules have complexity 2. © 2012 London Mathematical Society.
Sat, 01 Jun 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042682013-06-01T00:00:00Z
- Homomorphisms from specht modules to signed young permutation moduleshttps://scholarbank.nus.edu.sg/handle/10635/155269Title: Homomorphisms from specht modules to signed young permutation modules
Authors: Lim, KJ; Tan, KM
Abstract: © 2018, Institute of Mathematics. All rights reserved. We construct a class ΘR of homomorphisms from a Specht module (formula presented) to a signed permutation module Mℤ(α|β) which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any ϕ ∈ Homℤϭn (formula presented) lies in the -span of Θsstd, a subset of ΘR corresponding to semistandard λ-tableaux of type (α|β). We also study the conditions for which (formula presented) - a subset of HomFϭn (formula presented) induced by Θsstd - is linearly independent, and show that it is a basis for HomFϭn (formula presented) when Fϭn is semisimple.
Wed, 25 Apr 2018 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1552692018-04-25T00:00:00Z
- Jantzen filtration of Weyl modules, product of Young symmetrizers and denominator of Young's seminormal basishttps://scholarbank.nus.edu.sg/handle/10635/155231Title: Jantzen filtration of Weyl modules, product of Young symmetrizers and denominator of Young's seminormal basis
Authors: Fang, Ming; Lim, Kay Jin; Tan, Kai Meng
Abstract: Let $G$ be a connected reductive algebraic group over an algebraically closed
field of characteristic $p>0$, $\Delta(\lambda)$ denote the Weyl module of $G$
of highest weight $\lambda$ and $\iota_{\lambda,\mu}:\Delta(\lambda+\mu)\to
\Delta(\lambda)\otimes\Delta(\mu)$ be the canonical $G$-morphism. We study the
split condition for $\iota_{\lambda,\mu}$ over $\mathbb{Z}_{(p)}$, and apply
this as an approach to compare the Jantzen filtrations of the Weyl modules
$\Delta(\lambda)$ and $\Delta(\lambda+\mu)$. 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 is further characterized by the denominator of a
certain Young's seminormal basis vector in certain cases. We obtain explicit
formulas for the split condition in some cases.
Tue, 01 Jan 2019 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1552312019-01-01T00:00:00Z
- Periodic Lie moduleshttps://scholarbank.nus.edu.sg/handle/10635/155266Title: Periodic Lie modules
Authors: Lim, Kay Jin; Tan, Kai Meng
Abstract: © 2015 Elsevier Inc. Let p be a prime number and k be a positive integer not divisible by p. We describe the Heller translates of the periodic Lie module Lie(pk) in characteristic p and show that it has period 2p-2 when p is odd and 1 when p= 2. We also show these Lie modules are endo-. p-permutation modules.
Fri, 01 Jan 2016 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1552662016-01-01T00:00:00Z
- Asymptotic behaviour of Lie powers and Lie moduleshttps://scholarbank.nus.edu.sg/handle/10635/102892Title: Asymptotic behaviour of Lie powers and Lie modules
Authors: Bryant, R.M.; Lim, K.J.; Tan, K.M.
Abstract: Let V be a finite-dimensional FG-module, where F is a field of prime characteristic p and G is a group. We show that, when r is not a power of p, the Lie power L r(V) has a direct summand B r(V), which is a direct summand of the tensor power V ⊗r such that dim B r (V)/ dim L r (V) → 1 as r→∞. Similarly, for the same values of r, we obtain a projective submodule C(r) of the Lie module Lie (r) over F such that dim C(r)/dim Lie (r)→1 as r→∞. © 2011. Published by Oxford University Press. All rights reserved.
Sat, 01 Dec 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028922012-12-01T00:00:00Z
- Generic jordan type of the symmetric and exterior powershttps://scholarbank.nus.edu.sg/handle/10635/126629Title: Generic jordan type of the symmetric and exterior powers
Authors: Benson, D.J.; Lim, K.J.
Abstract: We prove a result relating the stable generic Jordan types of the symmetric and exterior powers of the Heller translations of a module for a finite elementary abelian p-group. In the case of the trivial module, the stable generic Jordan types of the symmetric and exterior powers of its Heller translations are completely described. © World Scientific Publishing Company.
Fri, 01 Aug 2014 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1266292014-08-01T00:00:00Z