Generalized Tesler matrices, virtual Hilbert series, and Macdonald polynomial operators
Abstrak
We generalize previous definitions of Tesler matrices to allow negative matrix entries and non-positive hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices. Our interpretation uses <i>virtual Hilbert series</i>, a new class of symmetric function specializations which are defined by their values on (modified) Macdonald polynomials. As a result of this interpretation, we obtain a Tesler matrix expression for the Hall inner product $\langle \Delta_f e_n, p_{1^{n}}\rangle$, where $\Delta_f$ is a symmetric function operator from the theory of diagonal harmonics. We use our Tesler matrix expression, along with various facts about Tesler matrices, to provide simple formulas for $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ and $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ involving $q; t$-binomial coefficients and ordered set partitions, respectively.
Topik & Kata Kunci
Penulis (1)
Andrew Timothy Wilson
Akses Cepat
- Tahun Terbit
- 2015
- Sumber Database
- DOAJ
- DOI
- 10.46298/dmtcs.2489
- Akses
- Open Access ✓