arXiv subjects
Publications and source records attributed to Kay Jin Lim.
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 $\mathrm{Lie}(pk)$ in characteristic $p$ and show that it has period $2p-2$ when $p$ is odd and $1$ when $p=2$.
We show that the Sylow $p$-subgroups of a symmetric group, respectively an alternating group, are characterized as the $p$-subgroups containing all elementary abelian $p$-subgroups up to conjugacy of the symmetric group, respectively the alternating group. We apply the characterization result for symmetric groups to compute the vertices of the hook Specht modules associated to the partition $(kp-p,1^p)$ under the assumption that $k \equiv 1$ mod $p$ and $k \not\equiv 1$ mod $p^2$.
We show that the simple modules of the Rouquier blocks of symmetric groups, in characteristic $p$ and having $p$-weight $w$ with $w < p$, have a common complexity $w$, and that when $p$ is odd, $D^{(p+1,1^{p-1})}$ has complexity 1, while the other simple modules labelled by a partition having $p$-weight 2 have complexity 2.
We show that the complexity of the Lie module $\mathrm{Lie}(n)$ in characteristic $p$ is bounded above by $m$ where $p^m$ is the largest $p$-power dividing $n$ and, if $n$ is not a $p$-power, is equal to the maximum of the complexities of $\Lie(p^i)$ for $1 \leq i \leq m$.
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 $p\geq 3$, 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 $S^{(p^p)}$ for $p\geq 3$.
Let $M$ be a left module for the Schur algebra $S(n,r)$, and let $s \in \mathbb{Z}^+$. Then $M^{\otimes s}$ is a $(S(n,rs), F\mathfrak{S}_s)$-bimodule, where the symmetric group $\mathfrak{S}_s$ on $s$ letters acts on the right by place permutations. We show that the Schur functor $f_{rs}$ sends $M^{\otimes s}$ to the $(F\mathfrak{S}_{rs},F\mathfrak{S}_s)$-bimodule $F\mathfrak{S}_{rs} \otimes_{F(\mathfrak{S}_r \wr \mathfrak{S}_s)} ((f_rM)^{\otimes s} \otimes F\mathfrak{S}_s)$. As a corollary, we obtain the effect of the Schur functor on the Lie power $L^s(M)$, symmetric power $S^s(M)$ and exterior power $\bigwedge^s(M)$ of $M$.
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 $V^{\otimes r}$ and which satisfies $\dim B^r(V)/\dim L^r(V) \to 1$ as $r \to \infty$. Similarly, for the same values of $r$, we obtain a projective submodule $C(r)$ of the Lie module $\Lie(r)$ over $F$ such that $\dim C(r)/\dim \Lie(r) \to 1$ as $r \to \infty$.
Let w be the p-weight of a hook partition μand E be an elementary abelian p-subgroup generated by w disjoint p-cycles. We compute the generic Jordan type of the Specht module corresponding to the partition μrestricted to E. In particular, we show that the complexity of the Specht module is precisely the p-weight of the partition μ.
We concern the VIGRE's conjecture; namely the complexity of a Specht module is the p-weight of the corresponding partition if and only if the partition is not p by p. In abelian defect case, we calculate the cohomological variety of the Specht modules. In particular, we show that the Specht modules have complexities exactly given by the p-weights of the corresponding partitions. For some p-regular partitions not more than p parts with empty p-core, we show that the Specht modules fit into the conjecture. For the partition p^p, we show that the Specht module has complexity p-1 and we study the rank variety of the Specht module restricted to a maximal elementary abelian p-subgroup of rank p.