Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p > 0, ∆(λ) denote the Weyl module of G of highest weight λ and ι λ,µ : ∆(λ+µ) → ∆(λ)⊗∆(µ) be the canonical G-morphism. We study the split condition for ι λ,µ over Z (p) , and apply this as an approach to compare the Jantzen filtrations of the Weyl modules ∆(λ) and ∆(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 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.
J. Carlson introduced the cohomological and rank variety for a module over a finite group algebra. We give a general form for the largest component of the variety for the Specht module for the partition (p p ) of p 2 restricted to a maximal elementary abelian p-subgroup of rank p. We determine the varieties of a large class of Specht modules corresponding to p-regular partitions. To any partition μ of np of not more than p parts with empty p-core we associate a unique partition Φ(μ) of np, where the rank variety of the restricted Specht module S μ ↓ En to a maximal elementary abelian p-subgroup E n of rank n is V En (k) if and only if V En (S Φ(μ) ) = V En (k). In some cases where Φ(μ) is a 2-part partition, we show that the rank variety V En (S μ ) is V En (k). In particular, the complexity of the Specht module S μ is n.
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. The elements of [λ] are called the nodes of [λ]. If m ∈ {0, 1, . . . , n} and if µ is a partition of m such that µ i λ i , for all i 1, then µ is called a subpartition of λ. One then has [µ] ⊆ [λ], and one calls [λ] [µ] a skew diagram.
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 $\mu$ is 2-regular, we show that the complexity of the Specht module $S^\mu$ is precisely the $p$-weight of the partition $\mu$. 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 ∈ Z + . Then M ⊗s is a (S(n, rs), F Ss)-bimodule, where the symmetric group Ss on s letters acts on the right by place permutations. We show that the Schur functor frs sends M ⊗s to the (F Srs, F Ss)-bimodule F Srs ⊗ F (Sr Ss) ((frM ) ⊗s ⊗F F Ss). 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 ). Mathematics Subject Classification (2010). Primary 20G43, 20C30.
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