A commutative algebra on degenerate CP^1 and Macdonald polynomials
By Boris Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, S. Yanagida. 2009
228 citations recorded by the source corpus.
We introduce a unital associative algebra A over degenerate CP^1. We show that A is a commutative algebra and whose Poincar'e series is given by the number of partitions. Thereby we can regard A as a smooth degeneration limit of the elliptic algebra introduced by one of the authors and Odesskii. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by one of the authors. It is found that the Ding-Iohara algebra provides us with an algebraic framework for the free filed construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting a la Babelon Bernard Billey, the Ruijsenaars difference operator and the operator M(q,t_1,t_2) of Okounkov-Pandharipande.
Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/f7b0405f2114c3f20491002afef30395ca75564e
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In Applied Mathematics
In Discrete Mathematics and Combinatorics
In Geometry and Topology
Citations in the corpus
- Equivariant K-theory of Hilbert schemes via shuffle algebra (2009, methodology): “The subalgebra generated by Ki is studied in [4]; in particular, Theorem 2.”
- The Shuffle Algebra Revisited (2012, methodology): “Proof This proposition was first stated and proved in the special case d = μ = 0 in [3], and we will generalize their idea in order to obtain the desired result.”