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66th Algebra DaySaturday, April 24, 2010
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66th Algebra DaySaturday, April 24, 2010
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Each semester the institute holds a day-long workshop in algebra
featuring speakers from Canada and abroad. The sixty-six Algebra Day
will take place on April 24, 2010 at the University of Ottawa.
Place: Room B005 (Dept. of Math. and Stat., 585 King Edward, Ottawa)
The 3rd Lie Day (April 23, KED B005)
9:15 coffee
9:30 - 10:30 Mikhail Khovanov (Columbia, USA) More about categorification of quantum groups.
10:45 - 11:45 Ivan Dimitrov (Queen's, Canada) Extreme components of the tensor product of modules over algebraic groups via Borel-Weil-Bott's theorem.
Abstract: In this talk I will explain how the celebrated theorem of Borel-Weil-Bott provides a natural realization of some extreme components of the tensor product of two irreducible modules of simple algebraic groups. I will concentrate mostly on the representation-theoretic and combinatorial side of the problem. I will also discuss a number of connections of our construction with problems coming from Representation Theory, Combinatorics, and Geometry, including questions about the Littlewood-Richardson cone related to Horn's conjecture, settled by Knutson and Tao in the late 1990's.
The talk is based on a joint work with Mike Roth.
12:00 - 13:00 Shulim Kaliman (Miami, USA) On the present state of Andersen-Lempert theory
Abstract: A smooth complex affine algebraic variety $X$ has the algebraic density property (ADP) if the Lie algebra generated by completely integrable algebraic vector fields coincides with the space of all algebraic vector fields on $X$. If $X$ is also equipped with a algebraic volume form (i.e. a non-vanishing section of the canonical bundle) then it has the algebraic volume density property (AVDP) when the similar fact holds for algebraic vector fields of zero divergence with respect to this form. These notions are the core of a new directions in complex analysis - the Andersen-Lempert theory. We establish ADP for a wide class of affine algebraic varieties that includes most of linear algebraic groups and homogeneous spaces, and AVDP for a class of objects that includes, in particular all Lie groups.
Lunch break
15:00 - 16:00 Jacques Hurtubise (McGill, Canada) Topology of moduli spaces
16:15 - 17:15 Francois Bergeron (UQAM, Canada) Diagonal harmonics (coinvariant spaces) of complex reflexion groups in several sets of variables
Abstract: For rank $n$ complex reflexion groups, we consider graded modules of diagonally harmonic polynomials in $k$ sets of variables on $n$ variables. We argue that their multigraded Hilbert series can be described in an uniform manner independent of $k$. These take the form of positive integer coefficients sums of complete homogeneous symmetric functions, indexed by partitions associated to each group element. These expressions specialize, in the case $k=1$ to the Poincar\'e polynomial of the group, and for $k=2$ to the the famous Garsia-Haiman modules, in the symmetric group case.
Dinner 18:30
Organizers: Erhard Neher and Kirill Zainoulline
