Shuffles and Schur positivity

• Applicant: Professor Dr. Ghislain Fourier
• Subject Area: Mathematics
• Term: from 2012 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 219422837

Schur functions are characters of simple sln-modules L(λ), they form a basis of the symmetric polynomials in n variables. Symmetric polynomials are called Schur positive (SP) if they are non-negative linear combinations of Schur functions. sλsμ – sλ’sμ’ being SP is equivalent to existence of a surjective map. For a fixed weight λ, a shu_ing operation on the set of pairs (sμ1, sμ2 ), s.t. μ1 + μ2 = λ, is introduced, "reducing" the difference between μ1 and μ2. It is conjectured that the difference between a pair and a shuffled pair is SP. Evidence is, among other, given by the fact, that the dimension of the associated tensor product increases for the shuffled pairs. For certain shuffles this conjecture was made by A. Okounkov et al, recently proven by T. Lam et al. There are several ideas how to prove the conjecture, a purely combinatorial way by giving explicit injection between highest weight vectors, a representation theoretical approach by finding generators and relations of associated fusion products (for current algebras), to name a few.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1388: Representation Theory