141 research outputs found
Stable branching rules for classical symmetric pairs
We approach the problem of obtaining branching rules from the point of view
of dual reductive pairs. Specifically, we obtain a stable branching rule for
each of 10 classical families of symmetric pairs. In each case, the branching
multiplicities are expressed in terms of Littlewood-Richardson coefficients.
Some of the formulas are classical and include, for example, Littlewood's
restriction rule as a special case.Comment: 26 page
Rawlsian Justice as the Core of a Game
It is suggested that the ethical notion of social contract can be formally modeled using the well-studied concept of the core of a game. This provides a mathematical technique for studying social contracts and theories of justice. The idea is applied to Rawlsian justice here
The Complex of Maximal Lattice Free Simplices
The simplicial complex K(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form {x : Ax
Global analysis by hidden symmetry
Hidden symmetry of a G'-space X is defined by an extension of the G'-action
on X to that of a group G containing G' as a subgroup. In this setting, we
study the relationship between the three objects:
(A) global analysis on X by using representations of G (hidden symmetry);
(B) global analysis on X by using representations of G';
(C) branching laws of representations of G when restricted to the subgroup
G'.
We explain a trick which transfers results for finite-dimensional
representations in the compact setting to those for infinite-dimensional
representations in the noncompact setting when is -spherical.
Applications to branching problems of unitary representations, and to spectral
analysis on pseudo-Riemannian locally symmetric spaces are also discussed.Comment: Special volume in honor of Roger Howe on the occasion of his 70th
birthda
Rawlsian Justice as the Core of a Game
It is suggested that the ethical notion of social contract can be formally modeled using the well-studied concept of the core of a game. This provides a mathematical technique for studying social contracts and theories of justice. The idea is applied to Rawlsian justice here.
Small representations of finite classical groups
Finite group theorists have established many formulas that express
interesting properties of a finite group in terms of sums of characters of the
group. An obstacle to applying these formulas is lack of control over the
dimensions of representations of the group. In particular, the representations
of small dimensions tend to contribute the largest terms to these sums, so a
systematic knowledge of these small representations could lead to proofs of
important conjectures which are currently out of reach. Despite the
classification by Lusztig of the irreducible representations of finite groups
of Lie type, it seems that this aspect remains obscure. In this note we develop
a language which seems to be adequate for the description of the "small"
representations of finite classical groups and puts in the forefront the notion
of rank of a representation. We describe a method, the "eta correspondence", to
construct small representations, and we conjecture that our construction is
exhaustive. We also give a strong estimate on the dimension of small
representations in terms of their rank. For the sake of clarity, in this note
we describe in detail only the case of the finite symplectic groups.Comment: 18 pages, 9 figures, accepted for publications in the proceedings of
the conference on the occasion of Roger Howe's 70th birthday (1-5 June 2015,
Yale University, New Haven, CT
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