The conjugacy action of $$S_n$$Sn and modules induced from centralisers

@article{Sundaram2016TheCA,
  title={The conjugacy action of \$\$S\_n\$\$Sn and modules induced from centralisers},
  author={S. Sundaram},
  journal={Journal of Algebraic Combinatorics},
  year={2016},
  volume={48},
  pages={179-225}
}
  • S. Sundaram
  • Published 18 March 2016
  • Mathematics
  • Journal of Algebraic Combinatorics
We establish, for the character table of the symmetric group, the positivity of the row sums indexed by irreducible characters, when restricted to various subsets of the conjugacy classes. A notable example is that of partitions with all parts odd. More generally, we study representations related to the conjugacy action of the symmetric group. These arise as sums of submodules induced from centraliser subgroups, and their Frobenius characteristics have elegant descriptions, often as a… 
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References

SHOWING 1-10 OF 41 REFERENCES
Some Aspects of Groups Acting on Finite Posets
Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type
Let G be a finite simple group of Lie type, and let πG be the permutation representation of G associated with the action of G on itself by conjugation. We prove that every irreducible complex
On the reduction of the conjugating representation of a finite group
Here we define the group vector y to be a g X l column vector whose entries are the elements of G, arranged so that the identity element 71 is first, and so that the hff elements of a class C9 of
On the Existence of Tableaux with Given Modular Major Index
We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case
C O ] 1 1 D ec 2 01 5 REPRESENTATION STABILITY FOR COHOMOLOGY OF CONFIGURATION SPACES IN R
This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group Sn on the cohomology of the configuration space of n ordered points in R. This
Theorem about the conjugacy representation ofSn
Every group has two natural representations on itself, the regular representation and the conjugacy representation. We know everything about the construction of the regular representation, but we
Representation stability for cohomology of configuration spaces in $\mathbf{R}^d$
This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group $S_n$ on the cohomology of the configuration space of $n$ ordered points in
The Representation Theory of the Symmetric Group
1. Symmetric groups and their young subgroups 2. Ordinary irreducible representations and characters of symmetric and alternating groups 3. Ordinary irreducible matrix representations of symmetric
Clifford theory and applications
This is an introduction to Clifford theory of induced representations from normal subgroups of finite groups. As an application, a complete explicit description of the irreducible representations of
Representations of finite and compact groups
Groups and counting principles Fundamentals of group representations Abstract theory of representations of finite groups Representations of concrete finite groups. I: Abelian and Clifford groups
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