ConferenceARTA III Advances in Representation Theory of Algebras
The third Advances in Representation Theory of Algebras (ARTA) meeting will be held at the Université du Québec à Montréal, Montréal, Québec, Canada, from the 16th to the 20th of June 2014.
Abstract: The Iwahori-Hecke algebra of the symmetric group is the convolution algebra arising from the variety of pairs of complete flags over a finite field. Considering convolution on the space of triples of two flags and a vector we obtain the mirabolic Hecke algebra, which had originally been described by Solomon. We will see a new presentation of this algebra which shows that it is a quotient of a cyclotomic Hecke algebra. This is similar to a result by Halverson and Ram on the q-rook algebra. With this new point of view, we can recover Siegel's results about the representations of the mirabolic Hecke algebra, as well as proving new 'mirabolic' analogues of classical results about the Iwahori-Hecke algebra.
We use the model of root-theoretic Young diagrams to establish combinatorial formulas for Schubert calculus of the (co)adjoint varieties.of classical Lie type, building on earlier Pieri-type rules of P.Pragacz-J.Ratajski and A.Buch-A.Kresch-H.Tamvakis. Using these rules, as well as results of P.E.Chaput-N.Perrin in the exceptional types, we suggest a connection between planarity of the diagrams and polytopality of the set of nonzero Schubert structure constants. (Joint with A. Yong). We also use root-theoretic Young diagrams to establish a new combinatorial formula for the GL_n Belkale-Kumar product (after A.Knutson-K.Purbhoo).