2009 Fiscal Year Final Research Report

Combinatorial structures of flag varieties and quantum deformation

Research Project

Project/Area Number	19740013
Research Category	Grant-in-Aid for Young Scientists (B)
Allocation Type	Single-year Grants
Research Field	Algebra
Research Institution	Kyoto University
Principal Investigator	MAENO Toshiaki Kyoto University, 大学院・工学研究科, 講師 (60291423)
Project Period (FY)	2007 – 2009
Keywords	旗多様体 / 鏡映群 / Hopf代数 / Lefschetz性
Research Abstract	In the present research, we revealed a relationship between a representation of the Nichols-Woronowicz algebra describing the braided differential structure on the Weyl groups and the elliptic Dunkl operators. We also constructed the Nichols-Woronowicz model of the coinvariant algebras of the complex reflection groups. Moreover, a new Pieri-type formula for the torus-equivariant (quantum) cohomology ring of the flag variety has been proved by means of an extension of the Fomin-Kirillov quadratic lgebra. In the direction of research on Lefschetz properties, we gave a characterization of he Lefschetz elements in finite-dimensional Gorenstein algebras in terms of the Hessians f the corresponding polynomials. As an application, we constructed a new series of orenstein algebras without the strong Lefschetz property.

(4 results)