DAHAs and Skein Theory

Higher-dimensional Heegaard Floer homology and Hecke algebras

“…At this point we observe that H 0 (Ω(UConf κ (Σ))) is isomorphic to the group algebra of the surface braid group of Σ over Z. In [MS21], Morton and Samuelson defined the braid skein algebra BSk κ (Σ), which is a quotient of the group algebra of the surface braid group over Z[s ±1 , c ±1 ] by the skein relation and the marked point relation; see Definition 4.1. Here s and c are parameters that appear in the skein and marked point relations, respectively.…”

“…(3) a topological description of DAHA of gl κ as a braid skein algebra due to Morton and Samuelson [MS21].…”

“…The main result from Morton-Samuelson [MS21] is that the double affine Hecke algebra Ḧκ of gl κ is naturally isomorphic to BSk κ (T 2 ). Hence we have:…”

Type $A$ DAHA and Doubly Periodic Tableaux

“…Application: proof of the Morton-Samuelson conjecture. In [15] the algebra Ḧq,t (m) was identified with the group algebra of braids on the torus T 2 \ { * } modulo certain relations. This algebra is then mapped to another algebra denoted by Sk n (T 2 , * ) (Theorem 4.1 in op.cit.).…”

“…This algebra is then mapped to another algebra denoted by Sk n (T 2 , * ) (Theorem 4.1 in op.cit.). By definition, Sk n (T 2 , * ) is an algebra over C[s ±1 , c ±1 , v ±1 ] consists of formal linear combinations of ribbon graphs in T 2 × I avoiding the so-called base string * × I modulo isotopy, skein relations and a relation that allows a band going on one side of the base string to be replaced by a band going on the other side multiplied by c 2 (see Section 4 in [15]). The bands have m inputs on T 2 × {0} and m outputs on T 2 × {1}.…”