465 research outputs found
Specht modules with abelian vertices
In this article, we consider indecomposable Specht modules with abelian
vertices. We show that the corresponding partitions are necessarily -cores
where is the characteristic of the underlying field. Furthermore, in the
case of , or and is 2-regular, we show that the complexity
of the Specht module is precisely the -weight of the partition
. 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 for
Straightening rule for an -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 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
, we discovered the number of compositions such that
can be rearranged to and whose partial sums of are not
divisible by
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.Comment: To appear in Adv. Math. 307 (2017) 369--416. Proposition 4.3 (F4),
(F5) corrected, Lemma 4.9 adjusted accordingl
A note on the signature representations of the symmetric groups
For a partition {\lambda} and a prime p, we prove a necessary and sufficient
condition for there exists a composition {\delta} such that {\delta} can be
obtained from {\lambda} after rearrangement and all the partial sums of
{\delta} are not divisible by p.Comment: This is a substantially revised version of previous ones. In the
third section, we calculate some explicit p-Kostka number
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
. 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.Comment: This expands the previous version by including extra sections (4 and
6
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