The irreducible representations (irreps) of the symmetric group \(S_n\) are parameterized by combinatorial objects called "Young diagrams", or shapes. A given irrep has a basis indexed by "Young tableaux" of that shape. One may construct a basis that consists of weight vectors (simultaneous eigenvectors) for a commutative subalgebra \(Y\) of the group algebra \({\mathbb {C}} S_n\).

The double affine Hecke algebra (DAHA) is a deformation of  the group algebra of the affine symmetric group enlarged by a (commutative) Laurent polynomial algebra \(Y\).

While it is quite difficult to parameterize all of the irreps of the DAHA, if we restrict our attention to those that have a basis of \(Y\)-weight vectors, we can. They are parameterized by certain skew  shapes, and the weight basis is  indexed by the "periodic tableaux" of that shape. In this talk, we will construct these irreps.

This is joint work with Takeshi Suzuki.

Note the unusual time!