Multiplicities of tensor eigenvalues

Eigenvalues, Multiplicities and Graphs

For a given graph, there is a natural question of the possible lists of multiplicities for the eigenvalues among the spectra of Hermitian matrices with that graph (no constraint is placed upon the diagonal entries of the matrices by the graph). Here, we survey some of what is known about this question and include some new information about it. There is a natural focus upon the case in which the...

Tensor Product Multiplicities

Multiplicities of Eigenvalues and Tree-Width of Graphs

Definition 1. OG is the set of real symmetric V_V matrices with entries ai, j such that ai, j<0 if [i, j] # E and ai, j=0 if i{ j and [i, j] E. An operator A # OG has a non-degenerate first eigenvalue *1 (groundstate) if G is connected (Perron and Frobenius). The invariant +(G) is defined using multiplicities of the second eigenvalue *2 for some real symmetric matrix A # OG . Moreover, +(G) is ...

Sharp Bounds for Eigenvalues and Multiplicities on Surfaces of Revolution

For surfaces of revolution diffeomorphic to S, it is proved that (S, can) provides sharp upper bounds for the multiplicities of all of the distinct eigenvalues. We also find sharp upper bounds for all the distinct eigenvalues and show that an infinite sequence of these eigenvalues are bounded above by those of (S, can). An example of such bounds for a metric with some negative curvature is pres...

Special Isogenies and Tensor Product Multiplicities

We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the corresponding complex semisimple Lie groups (or Lie algebras). We explain the inequalities in two ways: combinatorially, using Littelmann's Path Model, and geometrically, using isogenies between algebraic groups defined...