Induced sign representations and characters of Hecke algebras

introduction

• http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=55223&date=2012-04-30

• Many combinatorial formulas for computations in the symmetric group Sn can be modified appropriately to describe computations in the Hecke algebra Hn(q), a deformation of C[Sn].

induced sign characters

• Unfortunately, the known formulas for induced sign characters of Sn are not among these.
• For induced sign characters of Hn(q), we conjecture formulas which specialize at q=1 to formulas for induced sign characters of Sn.
• We will discuss evidence in favor of the conjecture, and relations to the chromatic quasi-symmetric functions of Shareshian and Wachs.

Given a partition \lambda=(\lambda_1,\cdots, \lambda_n) of n

1 define W_{\lambda}=S_{\lambda_1}\times S_{\lambda_2} \cdots \times S_{\lambda_k}

2 For each coset of the form wW_{\lambda},

define T_{wW_{\lambda}}=\sum_{v\in wW_{\lambda}}(-q)^{\ell(v)}T_{v}

If we set q=1, we get a sum looks like (\sum_{w\in W} w_{\lambda} sgn(v)v)

3 Let H_n(q) act by lefy multiplication on coset sums T_{D} where D is of the form wW_{\lambda}

4 this left multiplication can be expressed as matrix multiplication

Let \rho_{q}^{\lambda}(T_v)=matrix that correspondes to left multiplication by T_v.

Let \rho^{\lambda}(v)=matrix corresponding to left multiplication by v.

the trace/character associated to representation \rho_{q}^{\lambda} are usually denoted by \epsilon_{q}^{\lambda}

Q. What is a nice formula for \epsilon_{q}^{\lambda}(T_{v}) ? (open)