Triple clasp formulas for $C_2$ webs

Abstract

Using the light ladder basis for Kuperberg's C2 webs, we derive triple clasp formulas for idempotents projecting to the top summand in each tensor product of fundamental representations. We then find explicit formulas for the coefficients occurring in the clasps, by computing these coefficients as local intersection forms. Our formulas provide further evidence for Elias's clasp conjecture, which was given for type A webs, and suggests how to generalize the conjecture to non-simply laced types.

Triple Clasp Formulas for $\mathfrak{g}_2$

“…If we could show that the clasped web basis was a unitriangular change of basis away from the dual canonical basis, then using the present work it should be possible to argue that Conjecture 1.11 holds in the g 2 case when λ 3 = ̟. Similarly, the sl 3 case and the sp 4 case would follow from results in [8,2] when λ 3 = ̟.…”

“…a weight in the W orbit of ̟ i for some i) we write d ̟ to denote the minimal length element in w ∈ W so that w(̟) is dominant. 2 The formulas are given inductively by expanding one clasp with diagrams made up of lower-weight clasps. We use the maximum number of clasps in one diagram to name the formula.…”

“…We also expect there to be a more general form of the conjecture which describes what happens for µ ∈ V (̟) which are not in the extremal Weyl orbit. The work in this paper and [2] could give enough data to guess the answer when V (̟) µ is one dimensional, but we have not yet carried this out. We also believe the general form of the conjecture, i.e.…”

“…To generalize the Jones-Wenzl recursion to higher ranks, a first step is to find explicit formulas of clasps in the rank two cases. 2 In the sl 3 case, a quadruple clasp expansion was given by Ohtsuki and Yamada [26], where they called a clasp a magic element. Later, Dongseok Kim gave a single clasp expansion and a double clasp expansion for the sl 3 case as well [15,17].…”

“…In the sp 4 case, Kim gave single clasp expansions for the (a, 0)-clasp and (0, b)-clasp, and also deduced double clasp expansions for clasps with these weights [17]. However, an inductive formula for sp 4 (a, b)-clasp remained unknown until recently, when triple clasp expansions were given by the first named author in [2]. This also gave rise to a generalization of Elias's type A clasp conjecture to type C 2 .…”