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1. CMB Online first

Jantzen, Jens Carsten
 Maximal Weight Composition Factors for Weyl Modules Fix an irreducible (finite) root system $R$ and a choice of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group $G_k$ over $k$ with root system $k$. One associates to any dominant weight $\lambda$ for $R$ two $G_k$--modules with highest weight $\lambda$, the Weyl module $V (\lambda)_k$ and its simple quotient $L (\lambda)_k$. Let $\lambda$ and $\mu$ be dominant weights with $\mu \lt \lambda$ such that $\mu$ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists $k$ such that $L (\mu)_k$ is a composition factor of $V (\lambda)_k$, and they exhibit an example in type $E_8$ where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact, one that uses a classification of the possible pairs $(\lambda, \mu)$, and another one that relies only on the classification of root systems. Keywords:algebraic groups, represention theoryCategories:20G05, 20C20

2. CMB 2016 (vol 60 pp. 111)

Ghaani Farashahi, Arash
 Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space $L^2(G/H,\mu)$. Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formulaCategories:20G05, 43A85, 43A32, 43A40

3. CMB 2011 (vol 55 pp. 673)

Aizenbud, Avraham; Gourevitch, Dmitry
 Multiplicity Free Jacquet Modules Let $F$ be a non-Archimedean 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 groupCategories:20G05, 20C30, 20C33, 46F10, 47A67

4. CMB 2011 (vol 54 pp. 255)

Dehaye, Paul-Olivier
 On an Identity due to Bump and Diaconis, and Tracy and Widom A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener-Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump-Diaconis-Tracy-Widom identity is a differentiated version of the classical Jacobi-Trudi identity. Keywords:Toeplitz matrices, Jacobi-Trudi identity, SzegÅ limit theorem, Heine identity, Wiener-Hopf factorizationCategories:47B35, 05E05, 20G05

5. CMB 2000 (vol 43 pp. 79)

König, Steffen
 Cyclotomic Schur Algebras and Blocks of Cyclic Defect An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings. Categories:20G05, 20C20, 16G30, 17B37, 57M25
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