کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6414470 | 1630491 | 2015 | 30 صفحه PDF | دانلود رایگان |
A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two Sr-irreducibles into irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical basis theory, by first constructing a nonstandard Hecke algebra Br, which, though not a Hopf algebra, is a u-analogue of the Hopf algebra CSr in some sense (where u is the Hecke algebra parameter). For r=3, we study this Hopf-like structure in detail. We define a nonstandard Hecke algebra HË3(k)âH3âk, determine its irreducible representations over Q(u), and show that it has a presentation with a nonstandard braid relation that involves Chebyshev polynomials evaluated at 1u+uâ1. We generalize this to Hecke algebras of dihedral groups. We go on to show that these nonstandard Hecke algebras have bases similar to the Kazhdan-Lusztig basis of H3 and are cellular algebras in the sense of Graham and Lehrer.
Journal: Journal of Algebra - Volume 423, 1 February 2015, Pages 375-404