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Let $G$ denote the unramified quasi-split unitary group $\mathbb{U}(1,1)(F)$ over a $p$-adic field $F$ with residual characteristic $p \neq 2$. In this paper, we first construct a large family of irreducible representations of the maximal…

We consider quantum group representations for a semisimple algebraic group G at a complex root of unity q. Here q is allowed to be of any order. We revisit some fundamental results of Parshall-Wang and Andersen-Polo-Wen from the 90's. In…

In this note we consider representations of the group GL(n,F), where F is the field of real or complex numbers or, more generally, an arbitrary local field, in the space of equivariant line bundles over Grassmannians over the same field F.…

We study the space $Q_n$ of all configurations of $n$ ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for $n<6$ and describe its…

Consideration of a classification of the number of partitions of a natural number according to the members of sub-partitions differing from unity leads to a non-recursive formula for the number of irreducible representations of the…

We consider noncrossing partitions of [n] under the action of (i) the reflection group (of order 2), (ii) the rotation group (cyclic of order n) and (iii) the rotation/reflection group (dihedral of order 2n). First, we exhibit a bijection…

We study finite dimensional algebras that appear as fibers of quantum orders over a given point of variety of center. We present the formula for the number of irreducible representations and check it for it for the algebra of twisted…

We study unitary representations of semidirect products of a compact quantum group with a finite group. We give a classification of all irreducible unitary representations, a description of the conjugate representation of irreducible…

We develop general techniques for computing the fundamental group of the configuration space of $n$ identical particles, possessing a generic internal structure, moving on a manifold $M$. This group generalizes the $n$-string braid group of…

Let G be a split reductive algebraic group over a non-archimedean local field. We study the representation theory of a central extension $\G$ of G by a cyclic group of order n, under some mild tameness assumptions on n. In particular, we…

The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding of irreducible subquotients of a reducible principal series representation of $GL_n(F)$, $F$ a $p$-adic field (without specifying their multiplicities which is done…

Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group E6, and of its subgroups. We are therefore led…

A cover of a finite non-cyclic group $G$ is a family $\mathcal{H}$ of proper subgroups of $G$ whose union equals $G$. A cover of $G$ is called minimal if it has minimal size, and irredundant if it does not properly contain any other cover.…

We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set $\mathcal{O} \subset F_n$ which is contained in the union of…

We study the intermediate quantum groups $H_N\subset G\subset U_N^+$. The basic examples are $H_N,K_N,O_N,U_N,H_N^+,K_N^+,O_N^+,U_N^+$, which form a cube. Any other example $G$ sits inside the cube, and by using standard operations, namely…

We find the complete branching law for the restriction of certain unitary representations of $O(1,n+1)$ to the subgroups $O(1,m+1)\times O(n-m)$, $0\leq m\leq n$. The unitary representations we consider belong either to the unitary…

We perform the computations necessary to establish a multiplicity one statement for the irreducible representations of a finite spin group which in turn yields the classification of irreducible representations of finite spin groups. (The…

We consider a construction of the fundamental spin representations of the simple Lie algebras $\mathfrak{so}(n)$ in terms of binary arithmetic of fixed width integers. This gives the spin matrices as a Lie subalgebra of a…

Attached to any reductive Lie group $G$ is a "Cartan motion group" $G_0$ $-$ a Lie group with the same dimension as $G$, but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of…

In this paper we examine the saturation conjecture on decompositions of tensor products of irreducible representations for complex semisimple algebraic groups of type $D$ (the even \emph{spin} groups: Spin$(2n)$ for $n\ge 4$ an integer),…