# University of Toronto's Symplectic Geometry Seminar

Nov. 25 2002, 2:10pm

SS5017A

## John Millson

###
University of Maryland

##
The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra

** Abstract: **
In my lecture I will present joint work with Misha Kapovich
and Bernhard Leeb. I will begin by describing the GENERALIZED
TRIANGLE INEQUALITIES.
A geodesic segment in a symmetric space of noncompact type and rank m
is determined up to isometry by m real numbers
(not just the usual length). The generalized triangle inequalities are a
system of homogeneous linear inequalities that give conditions on 3
m-tuples
of real numbers that are necessary and sufficient in order that one can
assemble three geodesic segments with these parameters into a triangle.
It is a remarkable fact that the triangle inequalities play
a fundamental role in some important problems from algebra. Indeed,
given
a triple of dominant weights a,b,c (now we have 3 m-tuples of decreasing
INTEGERS), in order that the corresponding triple tensor product of
finite
dimensional irreducible representations contain the trivial
representation it
is necessary that the triple a,b,c satisfies the triangle inequalities.
It is a well-known recent theorem of Knutson and Tao
(the Saturation Conjecture for GL(m)) that the
triangle inequalities are also SUFFICIENT for the group GL(m). However
for
other groups they are no longer sufficient. I will present a conjecture
about what happens for other groups. The main evidence for the
conjecture is
that there is also a "Saturation Conjecture" for the (spherical)
Hecke algebra which I will prove (I will explain what this algebra is)
. There
is a very close connection between the two conjectures
and in fact they are equivalent for GL(m). From this equivalence
I will deduce a new proof of the theorem of Knutson and Tao.