14,054 research outputs found
The symplectic and algebraic geometry of Horn's problem
Horn's problem was the following: given two Hermitian matrices with known
spectra, what might be the eigenvalue spectrum of the sum? This linear algebra
problem is exactly of the sort to be approached with the methods of modern
Hamiltonian geometry (which were unavailable to Horn). The theorem linking
symplectic quotients and geometric invariant theory lets one also bring
algebraic geometry and representation theory into play. This expository note is
intended to elucidate these connections for linear algebraists, in the hope of
making it possible to recognize what sort of problems are likely to fall to the
same techniques that were used in proving Horn's conjecture.Comment: 16 pages, 1 figure; expository conference paper (second version has
inessential cosmetic changes
Schubert calculus and shifting of interval positroid varieties
Consider k x n matrices with rank conditions placed on intervals of columns.
The ranks that are actually achievable correspond naturally to upper triangular
partial permutation matrices, and we call the corresponding subvarieties of
Gr(k,n) the _interval positroid varieties_, as this class lies within the class
of positroid varieties studied in [Knutson-Lam-Speyer]. It includes Schubert
and opposite Schubert varieties, and their intersections, and is Grassmann dual
to the projection varieties of [Billey-Coskun].
Vakil's "geometric Littlewood-Richardson rule" [Vakil] uses certain
degenerations to positively compute the H^*-classes of Richardson varieties,
each summand recorded as a (2+1)-dimensional "checker game". We use his same
degenerations to positively compute the K_T-classes of interval positroid
varieties, each summand recorded more succinctly as a 2-dimensional "K-IP pipe
dream". In Vakil's restricted situation these IP pipe dreams biject very simply
to the puzzles of [Knutson-Tao].
We relate Vakil's degenerations to Erd\H os-Ko-Rado shifting, and include
results about computing "geometric shifts" of general T-invariant subvarieties
of Grassmannians.Comment: 35 pp; this subsumes and obviates the unpublished
http://arxiv.org/abs/1008.430
- …