Crystal bases and simple modules for Hecke algebras of type G(p,p,n)

2005/06/27 by Hu, Jun
#20C08 #20C20 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.math/0506555

Abstract

We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type G(p,p,n). This yields a classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [J. Hu, J. Algebra 267 (2003) 7-20]. The separated case was completed in [J. Hu, J. Algebra 274 (2004) 446--490]. Furthermore, we use Naito--Sagaki's work [S. Naito & D. Sagaki, J. Algebra 251 (2002) 461--474] on Lakshmibai--Seshadri paths fixed by diagram automorphisms to derive explicit formulas for the number of simple modules over these Hecke algebras. These formulas generalize earlier results of [M. Geck, Represent. Theory 4 (2000) 370-397] on the Hecke algebras of type D_n (i.e., of type G(2,2,n)).

Related

• Lectures on cyclotomic Hecke algebras
• Two maps on affine type A crystals and Hecke algebras
• Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras
• The number of simple modules for the Hecke algebras of type G(r,p,n) (with an appendix by Xiaoyi Cui)
• JMMO Fock space and Geck-Rouquier classification of simple modules for Hecke algebras