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Questions tagged [rt.representation-theory]

Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.

18
votes
5answers
1k views

How small can a group with an n-dimensional irreducible complex representation be?

More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex ...
20
votes
3answers
909 views

Why are Dynkin diagrams characterized by their eigenvalues?

The Dynkin diagrams An, Dn, E6, E7, E8 can be characterized among finite simple connected graphs by the property that their eigenvalues (that is, the eigenvalues of their adjacency matrices) all have ...
13
votes
1answer
605 views

What are the Schur functions of the eigenvalues of a non-negative integer matrix counting?

Let A be a non-negative integer square matrix with eigenvalues x1, x2, ... xn. Any symmetric function of these eigenvalues with integer matrices is an integer. I'm aware of the following results ...
3
votes
3answers
543 views

What is a formula for the “group-like Drinfeld element”?

Any quantized universal enveloping algebra (in fact, any toplogically quasi-triangular Hopf algebra) has an (in its completion) an element u called the Drinfeld element which gives an isomorphism from ...
4
votes
3answers
424 views

Functions on hyperbolic space and modular curves

The decomposition of $L^{2}\left(S^{2}\right)$ under $SO\left(3,\mathbb{R}\right)$ is well-known. Focus now on the hyperbolic plane $H$ presented as the quotient $SL\left(2,\mathbb{R}\right)/SO\left(...
5
votes
2answers
590 views

Are there interesting monoidal structures on representations of quantum affine algebras?

Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...
7
votes
3answers
3k views

Beilinson-Bernstein and Koszul duality

For geometric representation theorists down here. Consider the Beilinson-Bernstein theorem: Functor of global sections establishes the correspondence between twisted D-modules with fixed ...
4
votes
3answers
634 views

Questions about Quivers

Hi, The definition I have for a Path Algega of a quiver Q is that it is the algebra whose basis is formed by the oriented paths in Q, including the trivial ones. Apparently multiplication is given ...
3
votes
1answer
352 views

What is the “right” hermitian structure on tensor products of quantum group representations?

This is pretty specific, but there are some experts around. So, in Chari & Pressley, it's explained that in the standard *-structure, every irreducible, finite-dimensional representation of a ...
10
votes
1answer
2k views

How to understand character sheaves

There's a well-known series of articles by Lusztig about Character Sheaves. They have important connections to many things in (geometric) representation theory, e.g. 0904.1247 How to understand these ...
26
votes
18answers
16k views

Learning about Lie groups

Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.