Mathematics

It is shown that the endomorphism algebra of an arbitrary Young permutation module is cellular. Those are are quasi-hereditary are then determined.

For symmetric groups we show that the p-canonical basis can be extended to a cell datum for the Iwahori-Hecke algebra H and that the two-sided p-cell preorder coincides with the Kazhdan-Lusztig two-sided cell preorder. Moreover, we show that left (or right) p-cells inside the same two-sided p-cell for Hecke algebras of finite crystallographic Coxeter systems are incomparable (Property A).

Let Γ be a non-uniform lattice in SL(2,R) . In this paper, we study various L 2 -norms of automorphic representations of SL(2,R) . We will bound these norms with intrinsic norms defined on the representation. Comparison of these norms will help us understand the growth of L -functions in a systematic way.

Whittaker functions of GL(n,R) , are most known for its role in the Fourier-Whittaker expansion of cusp forms. Their behavior in the Siegel set, in large, is well-understood. In this paper, we insert into the literature some potentially useful properties of Whittaker function over the group GL(n,R) and the mirobolic group P n . We proved the square integrabilty of the Whittaker functions with respect to certain measures, extending a theorem of Jacquet and Shalika . For principal series representations, we gave various asymptotic bounds of smooth Whittaker functions over the whole group GL(n,R) . Due to the lack of good terminology, we use whittaker functions to refer to K -finite or smooth vectors in the Whittaker model.

We present a performant and rigorous algorithm for certifying that a matrix is close to being a projection onto an irreducible subspace of a given group representation. This addresses a problem arising when one seeks solutions to semi-definite programs (SDPs) with a group symmetry. Indeed, in this context, the dimension of the SDP can be significantly reduced if the irreducible representations of the group action are explicitly known. Rigorous numerical algorithms for decomposing a given group representation into irreps are known, but fairly expensive. To avoid this performance problem, existing software packages -- e.g. RepLAB, which motivated the present work -- use randomized heuristics. While these seem to work well in practice, the problem of to which extent the results can be trusted arises. Here, we provide rigorous guarantees applicable to finite and compact groups, as well as a software implementation that can interface with RepLAB. Under natural assumptions, a commonly used previous method due to Babai and Friedl runs in time O(n^5) for n-dimensional representations. In our approach, the complexity of running both the heuristic decomposition and the certification step is O(max{n^3 log n, D d^2 log d}), where d is the maximum dimension of an irreducible subrepresentation, and D is the time required to multiply elements of the group. A reference implementation interfacing with RepLAB is provided.

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a "large" family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even the isomorphism type of D is determined in this situation. Moreover, it can read off from the character table whether |D/D'|=4 where D' denotes the commutator subgroup of D. We also propose a new characterization of nilpotent blocks in terms of the character table.

We use the Darboux coordinate representation found by two of the authors (this http URL. and this http URL.) for entries of general symplectic leaves of the A n -groupoid of upper-triangular matrices to express roots of the characteristic equation det(A?��?A T )=0 , with A??A n , in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichmüller spaces for the algebra s l n . We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement remains valid in the quantum case as well. We consider a generalization of A n -groupoid to a A S p 2m -groupoid.

We determine the Verma multiplicities of standard filtrations of projective modules for integral atypical blocks in the BGG category O for the orthosymplectic Lie superalgebras osp(3|4) by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.

In this paper, we obtained character formulas of irreducible unitary representations of U(n,n+1) by using Howe's correspondence and the Cauchy--Harish-Chandra integral. The representations of U(n,n+1) we are dealing with are obtained from a double lifting of a representation of U(1) via the dual pairs (U(1),U(1,1)) and (U(1,1),U(n,n+1)) .

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers.