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Signed two-direction multiedge quiver-flag crossing

Example

Assume the Axiom of Choice as inherited. Take the loopless quiver with two vertices $i, j$ and arrow multiplicities $h_{i j} = 2$ , $h_{j i} = 1$ , the dimension vector $β = 2 α_{i} + α_{j}$ and the word $ν = (i, j, i)$ . Then $Q_{i j} (u, v) = (u - v)^{3}$ , and the dimension differences are $d_{s_{1} ν} - d_{ν} = h_{i j} - h_{j i} = 1$ and $d_{s_{2} ν} - d_{ν} = h_{j i} - h_{i j} = - 1$ . The two triple products of the native braid relation act on the $ν$ -summand by $τ_{2} τ_{1} τ_{2} e_{ν} = (x_{2} - x_{3})^{2} s_{2} D_{1} (x_{2} - x_{3}) s_{2} e_{ν}, τ_{1} τ_{2} τ_{1} e_{ν} = (x_{1} - x_{2}) s_{1} D_{2} (x_{1} - x_{2})^{2} s_{1} e_{ν},$ where $D_{1} = (1 - s_{1}) / (x_{1} - x_{2})$ acts on the word $(i, i, j)$ and $D_{2} = (1 - s_{2}) / (x_{2} - x_{3})$ acts on the word $(j, i, i)$ , while the distinct-color crossings act by $(x_{1} - x_{2})^{2} s_{1}$ or $(x_{2} - x_{3}) s_{2}$ on $(i, j, i)$ and by $(x_{2} - x_{3})^{2} s_{2}$ or $(x_{1} - x_{2}) s_{1}$ on the intermediate words. Since $ν_{1} = ν_{3} = i$ , their difference equals the native correction $\frac{Q_{i j} (x_{1}, x_{2}) - Q_{i j} (x_{3}, x_{2})}{x_{3} - x_{1}} = - [(x_{1} - x_{2})^{2} + (x_{1} - x_{2}) (x_{3} - x_{2}) + (x_{3} - x_{2})^{2}],$ a nonzero polynomial whenever $h_{i j} + h_{j i} \geq 1$ . Both oriented crossing factors $(x_{1} - x_{2})^{2}$ and $(x_{2} - x_{3})$ occur, so the multiedge multiplicities and both arrow directions enter the correction.

Facts & Assumptions

Given: the Axiom of Choice; the two-vertex quiver with $h_{i j} = 2$ , $h_{j i} = 1$ ; the dimension vector $β = 2 α_{i} + α_{j}$ ; the word $ν = (i, j, i)$ ; the incidence data, generators and presentation of Quiver full-flag incidence varieties and signed geometric KLR generators.

[F1]

Assume AC. Quiver full-flag incidence varieties and signed geometric KLR generators fixes $d_{ν} = \sum_{i} β_{i} (β_{i} - 1) / 2 + \sum_{a > b} h_{ν_{a} ν_{b}}$ , the normalized elementary generators $σ_{r}, τ_{r}$ with $τ_{r} e_{ν} = (- 1)^{h_{ν_{r} ν_{r + 1}}} σ_{r} e_{ν}$ for distinct colors, $a_{i j} = - h_{i j} - h_{j i}$ , $Q_{i j} (u, v) = (- 1)^{h_{i j}} (u - v)^{h_{i j} + h_{j i}}$ and the braid correction $[Q_{ν_{r} ν_{r + 1}} (x_{r}, x_{r + 1}) - Q_{ν_{r} ν_{r + 1}} (x_{r + 2}, x_{r + 1})] / (x_{r + 2} - x_{r})$ when $ν_{r} = ν_{r + 2}$ .

[F2]

Assume AC. Quiver-flag convolution generators satisfy the signed KLR relations supplies the action formulas: $τ_{r} (f) = (x_{r} - x_{r + 1})^{h_{i j}} s_{r} (f)$ for distinct colors $i = ν_{r} \neq j = ν_{r + 1}$ and $τ_{r} (f) = (f - s_{r} f) / (x_{r} - x_{r + 1})$ for equal colors, with $s_{r}$ swapping both the two variables and the two letters.

[A1]

The Axiom of Choice is present only because the consumed definition and relation suppliers inherit it; the displayed polynomial expansion is finite and choice-free (The Axiom of Choice).

Verification

technique · direct

1.1F1given

Dimensions and normalizations. With $h_{i j} = 2$ , $h_{j i} = 1$ the defining polynomial is $Q_{i j} (u, v) = (- 1)^{2} (u - v)^{3} = (u - v)^{3}$ , and the dimension formula gives $d_{(i, j, i)} = 1 + (h_{j i} + h_{i j}) = 4$ , $d_{(j, i, i)} = 1 + 2 h_{i j} = 5$ and $d_{(i, i, j)} = 1 + 2 h_{j i} = 3$ ; hence $d_{s_{1} ν} - d_{ν} = 1$ and $d_{s_{2} ν} - d_{ν} = - 1$ . The unequal crossing normalization exponent $d_{s_{r} ν} - \dim_{C} C_{s_{r} ν, ν} = h_{ν_{r} ν_{r + 1}}$ is instead $2$ for $r = 1$ and $1$ for $r = 2$ ; it must not be identified with the dimension difference $d_{s_{r} ν} - d_{ν}$ .

2.1F2step 1.1

The crossing operators. The action formulas of [F2] give, on the relevant words, $τ_{1} e_{(i, j, i)} = (x_{1} - x_{2})^{2} s_{1}$ , $τ_{2} e_{(i, j, i)} = (x_{2} - x_{3}) s_{2}$ , $τ_{2} e_{(i, i, j)} = (x_{2} - x_{3})^{2} s_{2}$ , $τ_{1} e_{(j, i, i)} = (x_{1} - x_{2}) s_{1}$ , together with the equal-color divided differences $D_{1}$ on $(i, i, j)$ and $D_{2}$ on $(j, i, i)$ ; this exhibits the two triple products as the displayed composite operators on $k [x_{1}, x_{2}, x_{3}]$ .

3.1A1F1F2step 2.1algebra∎

Put $x = x_{1}$ , $y = x_{2}$ , $z = x_{3}$ and $w_{0} = s_{2} s_{1} s_{2} = s_{1} s_{2} s_{1}$ . In the skew group algebra the two composite operators of step 2.1 expand to $A = [(y - z)^{2} (z - y) - (y - z)^{2} (x - y) w_{0}] / (x - z)$ and $B = [(x - y) (y - x)^{2} - (x - y) (y - z)^{2} w_{0}] / (x - z)$ . Their $w_{0}$ coefficients coincide. Thus $A - B = ((z - y)^{3} - (x - y)^{3}) / (x - z) = - [(x - y)^{2} + (x - y) (z - y) + (z - y)^{2}]$ , exactly the native correction. It is nonzero of total polynomial degree $2$ , since its $x^{2}$ coefficient is $- 1$ . This explicit expansion retains both the square factor from $h_{i j} = 2$ and the linear factor from $h_{j i} = 1$ ; all divisions represent polynomial operators by [F2].

Depends on

Used by

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Sources

M. Varagnolo, E. Vasserot, Canonical bases and KLR-algebras, arXiv:0901.3992; complete 30-page article retrieved and read at the cited sections. (standard reference, not scraped)
R. Rouquier, Quiver Hecke algebras and 2-Lie algebras, arXiv:1112.3619; complete 48-page article retrieved and read at the cited sections. (standard reference, not scraped)