Joakim Uhlin: Combinatorics of MacDonald polynomials and cyclic sieving

Tid: On 2019-01-23 kl 10.15 - 11.15

Föreläsare: Joakim Uhlin (Master student)

Plats: Room 3418, math department KTH, floor 4 (bottom floor)

In this thesis, we study the non-symmetric Macdonald polynomials Eλ(x; q, t) at t = 0 from a combinatorial point of view, using the combinatorial formula found by J. Haglund, M. Haiman, and N. Loehr. Our primary focus is when λ is a partition. We summarize some known theory about this specialization and prove some new results related to this combinatorial formula.
We also define the cyclic sieving phenomenon (CSP). For rectangular λ, we present an instance of cyclic sieving with Eλ(1, q, q^2 . . . , q^(k−1); 1, 0) as CSP-polynomial. We also conjecture another instance of CSP with Eλ(1,1,...,1;q,0) as CSP-polynomial. This conjecture generalizes a previously known CSP-triple. Furthermore, we prove this conjecture in the case when λ is an m × 2 diagram.