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For any reductive group G and a parabolic subgroup P with its Levi subgroup L , the first author in [5] introduced a ring homomorphism ξ λ P : Rep λ − poly C (L) → H ⁎ (G / P , C) , where Rep λ − poly C (L) is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation V (λ) of G with highest weight λ). In this paper we study this homomorphism for G = Sp (2 n) and its maximal parabolic subgroups P n − k for any 1 ≤ k ≤ n − 1 (with the choice of V (λ) to be the defining representation V (ω 1) in C 2 n). Thus, we obtain a C -algebra homomorphism ξ n , k : Rep ω 1 − poly C (Sp (2 k)) → H ⁎ (I G (n − k , 2 n) , C). Our main result asserts that ξ n , k is injective when n tends to ∞ keeping k fixed. Similar results are obtained for the odd orthogonal groups. [ABSTRACT FROM AUTHOR]

The paper consists of two parts. The first part introduces the representation ring for the family of compact unitary groups U ( 1 ) , U ( 2 ) , … . This novel object is a commutative graded algebra R with infinite-dimensional homogeneous components. It plays the role of the algebra of symmetric functions, which serves as the representation ring for the family of finite symmetric groups. The purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper. The second part deals with a family of Markov processes on the dual object to the infinite-dimensional unitary group U ( ∞ ) . These processes were defined in a joint work with Alexei Borodin (2012) [5] . The main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators. It is shown that the generators are implemented by certain second order partial differential operators with countably many variables, initially defined as operators on R . [ABSTRACT FROM AUTHOR]

Denote by Q the generalized quaternion group of order 4 m. Let R( Q) be its complex representation ring, and Δ( Q) its augmentation ideal. In this paper, the author gives an explicit Z-basis for the Δ( Q) and determines the isomorphism class of the n-th augmentation quotient $$\frac{{{\Delta ^n}\left( {{Q_m}} \right)}}{{{\Delta ^{n + 1}}\left( {{Q_m}} \right)}}$$ for each positive integer n. [ABSTRACT FROM AUTHOR]

Abstract: In the theory of canonical induction formulae for Mackey functors, Boltje [4] demonstrated that the plus constructions, together with the mark morphism, are useful for the study of canonical versions of induction theorems analogous to those in representation theory of finite groups. In this paper, we present a short exact sequence for the plus constructions derived from Cauchy–Frobenius lemma, and apply it to the proof of Boltje's integrality result for canonical induction formulae. The methods appearing in Boltje's theory, combined with the Dress construction for Mackey functors, are applicable to induction theorems on representations of the twisted quantum double of a finite group. As a sequel to such a research, we describe canonical versions of two induction theorems whose origins are Artin's induction theorem and Brauer's induction theorem on -characters of a finite group. [Copyright &y& Elsevier]

In topology there is a well-known theorem of Atiyah, Hirzebruch, and Segal which states that for a connected compact Lie group G there is an isomorphism R(G) ≅ K0(BG), where BG is the classifying space of G. In the present paper we consider an algebraic analogue of this theorem. For a split reductive group G over a field k, we prove that there is a natural isomorphism ... where KnG is Thomason's G-equivariant K-theory of Spec k, BG is a motivic étale classifying space introduced by Voevodsky and Morel, and IG s the augmentation ideal of K0G (k). [ABSTRACT FROM AUTHOR]

Abstract: The direct sum and tensor product (defined point-wise and arrow-wise) of representations of a given quiver is encoded in the representation ring of that quiver. We provide methods which reduce the computation of certain parts of the representation ring of a quiver to that of connected subquivers. These methods, combined with known results on the representation rings of quivers of type and , and supplemented by certain matrix calculations, yield an explicit description of the representation rings of all quivers of type . [Copyright &y& Elsevier]

Abstract: Rhetorical biset functors can be defined for any family of finite groups that is closed under subquotients up to isomorphism. The rhetorical p-biset functors almost coincide with the rational p-biset functors. We show that, over a field with characteristic zero, the rhetorical biset functors are semisimple and, furthermore, they admit a character theory involving primitive characters of automorphism groups of cyclic groups. [Copyright &y& Elsevier]

The ring of monomial representations of a finite group has been investigated by Dress (1971) and Boltje (1990), among others. It is of interest in connection with induction theorems in representation theory. Its species have recently been determined by Boltje. In this article, we will analyze the block distribution of species. As an application, we will determine the prime ideals of the ring of monomial representations. The results here constitute a slightly modified version of part of the first author's Diplomarbeit (Fotsing, 2003), written under the direction of the second author. [ABSTRACT FROM AUTHOR]

We solve the Clebsch–Gordan problem for rep핂 Q (i.e. we describe the Krull–Schmidt decomposition of A ⊗ B for all A, B ∈ rep핂 Q) in case 핂 is an algebraically closed field of characteristic 0 and Q is a quiver of type $\tilde{\mathbb{A}}_n$. The underlying tensor product is defined point-wise and arrow-wise. [ABSTRACT FROM AUTHOR]

Abstract: We present some results about the Dress construction of Green functor associated to a G-monoid. In particular, we see that the crossed Burnside rings, the representation rings of quantum double and Hochschild cohomology rings are constructed by the Dress construction of some Green functors. [Copyright &y& Elsevier]