1. CMB 2016 (vol 60 pp. 154)
 Liu, Ye

On Chromatic Functors and Stable Partitions of Graphs
The chromatic functor of a simple graph is a functorization of
the chromatic polynomial. M. Yoshinaga showed
that two finite graphs have isomorphic chromatic functors if
and only if they have the same chromatic polynomial. The key
ingredient in the proof is the use of stable partitions of graphs.
The latter is shown to be closely related to chromatic functors.
In this note, we further investigate some interesting properties
of chromatic functors associated to simple graphs using stable
partitions. Our first result is the determination of the group
of natural automorphisms of the chromatic functor, which is in
general a larger group than the automorphism group of the graph.
The second result is that the composition of the chromatic functor
associated to a finite graph restricted to the category $\mathrm{FI}$
of finite sets and injections with the free functor into the
category of complex vector spaces yields a consistent sequence
of representations of symmetric groups which is representation
stable in the sense of ChurchFarb.
Keywords:chromatic functor, stable partition, representation stability Categories:05C15, 20C30 

2. CMB 2011 (vol 55 pp. 673)
 Aizenbud, Avraham; Gourevitch, Dmitry

Multiplicity Free Jacquet Modules
Let $F$ be a nonArchimedean local field or a finite field.
Let $n$ be a natural number and $k$ be $1$ or $2$.
Consider $G:=\operatorname{GL}_{n+k}(F)$ and let
$M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup.
Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.
Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$.
In this paper we prove that $J$ is a multiplicity free functor, i.e.,
$\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$,
for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$.
We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors.
At the end we discuss whether other Jacquet functors are multiplicity free.
Keywords:multiplicity one, Gelfand pair, invariant distribution, finite group Categories:20G05, 20C30, 20C33, 46F10, 47A67 

3. CMB 2011 (vol 54 pp. 297)
 Johnson, Marianne; Stöhr, Ralph

Lie Powers and PseudoIdempotents
We give a new factorisation of the classical Dynkin operator,
an element of the integral group ring of the symmetric group that
facilitates projections of tensor powers onto Lie powers.
As an application we show that the iterated Lie power $L_2(L_n)$ is
a module direct summand of the Lie power $L_{2n}$ whenever the
characteristic of the ground field does not divide $n$. An explicit
projection of the latter onto the former is exhibited in this case.
Categories:17B01, 20C30 

4. CMB 2009 (vol 52 pp. 9)
 Chassé, Dominique; SaintAubin, Yvan

On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics
Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements
$[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the
symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{1})}$ with
$0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in
\sn$. We give the spectrum of $R_n$ and show that the ratio of the
largest eigenvalue $\lambda_0$ to the second largest one (in absolute
value) increases as a positive power of $n$ as $n\rightarrow \infty$.
Keywords:symmetric group, representation theory, eigenvalue, statistical physics Categories:20B30, 20C30, 15A18, 82B20, 82B28 

5. CMB 2008 (vol 51 pp. 584)
 Purbhoo, Kevin; Willigenburg, Stephanie van

On Tensor Products of Polynomial Representations
We determine the necessary and sufficient combinatorial
conditions for which the tensor product of two irreducible polynomial
representations of $\GL(n,\mathbb{C})$ is isomorphic to another.
As a consequence we discover families of LittlewoodRichardson
coefficients that are nonzero, and a condition on Schur nonnegativity.
Keywords:polynomial representation, symmetric function, LittlewoodRichardson coefficient, Schur nonnegative Categories:05E05, 05E10, 20C30 
