Graphical Calculus for the Double Affine Q-Dependent Braid Group

Abstract

We define a double affine $Q$-dependent braid group. This group is constructed by appending to the braid group a set of operators $Q_i$, before extending it to an affine $Q$-dependent braid group. We show specifically that the elliptic braid group and the double affine Hecke algebra (DAHA) can be obtained as quotient groups. Complementing this we present a pictorial representation of the double affine $Q$-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation we can fully describe any DAHA. Specifically, we graphically describe the parameter $q$ upon which this algebra is dependent and show that in this particular representation $q$ corresponds to a twist in the ribbon.

References in corpus (2)

• On polynomials interpolating between the stationary state of a O(n) model and a Q.H.E. ground state
• Highest weight Macdonald and Jack Polynomials

Cited by in corpus (1)

• Spin-Ruijsenaars, q-deformed Haldane-Shastry and Macdonald polynomials