Abacus-Tournament Models of Hall-Littlewood Polynomials
Wills, Andrew Johan
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In this dissertation, we introduce combinatorial interpretations for three types of Hall- Littlewood polynomials (denoted R[BULLET], P[BULLET], and Q[BULLET]) by using weighted combinatorial objects called abacus-tournaments. We then apply these models to give combinatorial proofs of properties of Hall-Littlewood polynomials. For example, we show why various specializations of Hall-Littlewood polynomials produce the Schur symmetric polynomials, the elementary symmetric polynomials, or the t-analogue of factorials. With the abacus-tournament model, we give a bijective proof of a Pieri rule for Hall-Littlewood polynomials that gives the P[BULLET]-expansion of the product of a Hall-Littlewood polynomial P[BULLET] with an elementary symmetric polynomial. We also give a bijective proof of certain cases of a second Pieri rule that gives the P[BULLET]-expansion of the product of a Hall-Littlewood polynomial P[BULLET] with another Hall-Littlewood polynomial Q(r). In general, proofs using abacus-tournaments focus on canceling abacus-tournaments and then [BULLET]finding weight-preserving bijections between the sets of uncanceled abacus-tournaments.
- Doctoral Dissertations