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The two-vertex KLR polynomial representation

Example

Take $I = {i, j}$ , $n = 2$ and the KLR datum with $Q_{i j} = 1$ if $a_{i j} = 0$ and $Q_{i j} (u, v) = u + v$ if $a_{i j} = a_{j i} = - 1$ (simply-laced edge, hence $d_{i} = d_{j}$ ), $Q_{j i}$ determined by (Q2). Then $I^{α_{i} + α_{j}} = {i j, j i}$ and the PBW basis of $R (α_{i} + α_{j})$ is ${x_{1}^{a} x_{2}^{b} τ_{1}^{ε} e (ν) : ε \in {0, 1}, ν \in {i j, j i}}$ , so $R (α_{i} + α_{j})$ is free of rank $4$ over $k [x_{1}, x_{2}]$ (and rank $2$ over the direct-sum polynomial algebra with its two idempotents) with graded dimension $\frac{2 (1 + q^{- b_{i j}})}{(1 - q^{2 d_{i}}) (1 - q^{2 d_{j}})}$ . (a) $a_{i j} = 0$ : $τ_{1}^{2} = e (i j) + e (j i)$ and $τ_{1} x_{1} = x_{2} τ_{1}$ and $τ_{1} x_{2} = x_{1} τ_{1}$ ; right multiplication by $τ_{1}$ gives a degree-zero isomorphism $R (α_{i} + α_{j}) e (i j) ≅ R (α_{i} + α_{j}) e (j i)$ of graded left modules, hence $[P_{i j}] = [P_{j i}]$ in $K_{0} (R)$ — the categorical form of $θ_{i} θ_{j} = θ_{j} θ_{i}$ . (b) $a_{i j} = - 1$ : choose the factorization $P_{i j} (u, v) = u + v$ , $P_{j i} (u, v) = 1$ ; the polynomial action of Faithful polynomial representation of a KLR algebra is $τ_{1} (f e (i j)) = (x_{1} + x_{2}) f (x_{2}, x_{1}) e (j i), τ_{1} (f e (j i)) = f (x_{2}, x_{1}) e (i j),$ so $τ_{1}^{2} = (x_{1} + x_{2})$ on both components, i.e. $τ_{1}^{2} e (ν) = Q_{ν_{1} ν_{2}} (x_{1}, x_{2}) e (ν)$ ; the algebra is free of rank $2$ over the two-idempotent polynomial algebra $k [x_{1}, x_{2}] e (i j) \oplus k [x_{1}, x_{2}] e (j i)$ (rank $4$ over the bare $k [x_{1}, x_{2}]$ ), and the three-strand divided-power classes satisfy $[P_{i j i}] = [P_{i (2) j}] + [P_{j i (2)}]$ in $K_{0} (R)$ , the simply-laced Serre relation at $a_{i j} = - 1$ .

Facts & Assumptions

Given: The two-vertex KLR datum with $a_{i j} = a_{j i} \in {0, - 1}$ , $d_{i}, d_{j}$ (equal on an edge), and the algebra $R (α_{i} + α_{j})$ of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum with the polynomial representation of Faithful polynomial representation of a KLR algebra and the PBW basis of The PBW (monomial) basis theorem for KLR algebras.

[F1]

Relations (4) and (6) of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum: $τ_{1}^{2} e (ν) = Q_{ν_{1} ν_{2}} (x_{1}, x_{2}) e (ν)$ and the dot slides, with $τ_{1}^{2} = Q_{i j}$ read on $e (i j)$ and $Q_{j i}$ on $e (j i)$ .

[F2]

Polynomial action (Faithful polynomial representation of a KLR algebra): with a factorization $Q_{i j} (u, v) = P_{i j} (u, v) P_{j i} (v, u)$ the action on unequal colours is $τ_{1} (f e (ν)) = P_{ν_{1} ν_{2}} (x_{2}, x_{1}) s_{1} (f) e (s_{1} ν)$ , and this action is faithful; a factorization with both factors nonzero exists.

[F3]

PBW basis (The PBW (monomial) basis theorem for KLR algebras): the monomials $τ_{1}^{ε} x_{1}^{a} x_{2}^{b} e (ν)$ , $ε \in {0, 1}$ , $ν \in {i j, j i}$ , form a $k$ -basis of $R (α_{i} + α_{j})$ .

[F4]

Serre relation for the Grothendieck ring (The KLR Grothendieck ring satisfies the divided-power quantum Serre relations): in $K_{0} (R)$ the divided classes satisfy $\sum_{a + b = - a_{i j} + 1} (- 1)^{a} [P_{i, a}] [P_{j, 1}] [P_{i, b}] = 0$ , which at $a_{i j} = - 1$ reads $[P_{i j i}] = [P_{i (2) j}] + [P_{j i (2)}]$ .

Verification

technique · direct; read off the two-strand PBW basis and check both parameter cases

1.1givenF3algebra

(Two-strand basis and dimension.) In weight $α_{i} + α_{j}$ the sequences are $i j$ and $j i$ , and $S_{2} = {1, s_{1}}$ , so the PBW basis [F3] consists of the four families $x_{1}^{a} x_{2}^{b} e (i j)$ , $x_{1}^{a} x_{2}^{b} τ_{1} e (i j)$ , $x_{1}^{a} x_{2}^{b} e (j i)$ , $x_{1}^{a} x_{2}^{b} τ_{1} e (j i)$ , which is rank $4$ over $k [x_{1}, x_{2}]$ ; grouping by idempotent gives rank $2$ over $k [x_{1}, x_{2}] e (i j) \oplus k [x_{1}, x_{2}] e (j i)$ . The graded dimension is the two geometric series for the dots in degrees $2 d_{i}, 2 d_{j}$ in each corner plus the crossing contributions: in the $i j$ -corner, $τ_{1} e (i j)$ has degree $- b_{i j}$ , and in the $j i$ -corner $τ_{1} e (j i)$ has degree $- b_{j i} = - b_{i j}$ , giving $\frac{1 + q^{- b_{i j}}}{(1 - q^{2 d_{i}}) (1 - q^{2 d_{j}})}$ in each corner, total $\frac{2 (1 + q^{- b_{i j}})}{(1 - q^{2 d_{i}}) (1 - q^{2 d_{j}})}$ .

1.2givenF1algebra

(Case $a_{i j} = 0$ .) Here $Q_{i j} = 1$ is a nonzero constant with $Q_{j i} = 1$ , so [F1] gives $τ_{1}^{2} e (i j) = e (i j)$ and $τ_{1}^{2} e (j i) = e (j i)$ , i.e. $τ_{1}^{2} = e (i j) + e (j i)$ , and the dot slides $τ_{1} x_{1} = x_{2} τ_{1}$ , $τ_{1} x_{2} = x_{1} τ_{1}$ (distinct colours). Right multiplication by $τ_{1}$ is an isomorphism of graded left modules $R (α_{i} + α_{j}) e (i j) \to R (α_{i} + α_{j}) e (j i)$ with inverse right multiplication by $τ_{1}$ on the other side: both composites are $τ_{1}^{2} = i d$ on the respective corners, and the map is degree-zero because $\deg τ_{1} = - b_{i j} = 0$ . Hence $[P_{i j}] = [R (α_{i} + α_{j}) e (i j)] = [R (α_{i} + α_{j}) e (j i)] = [P_{j i}]$ in $K_{0} (R)$ .

2.1step 1.1F2F4algebra∎

(Case $a_{i j} = - 1$ .) With $P_{i j} = u + v$ , $P_{j i} = 1$ we have $P_{i j} (u, v) P_{j i} (v, u) = (u + v) \cdot 1 = Q_{i j} (u, v)$ , a valid factorization with both factors nonzero, so [F2] applies: $τ_{1} (f e (i j)) = (x_{1} + x_{2}) f (x_{2}, x_{1}) e (j i)$ and $τ_{1} (f e (j i)) = 1 \cdot f (x_{2}, x_{1}) e (i j)$ . Applying twice, $τ_{1}^{2} (f e (i j)) = (x_{1} + x_{2}) f$ and $τ_{1}^{2} (f e (j i)) = (x_{1} + x_{2}) f$ , so $τ_{1}^{2} e (ν) = Q_{ν_{1} ν_{2}} (x_{1}, x_{2}) e (ν)$ on both summands, as required by [F1]. The freeness statements are step 1.1. For the Serre relation, apply [F4] with $a_{i j} = - 1$ : the sum runs over $a + b = 2$ with signs $+, -, +$ , i.e. $[P_{i, 2}] [P_{j}] - [P_{i, 1}] [P_{j}] [P_{i, 1}] + [P_{j}] [P_{i, 2}] = 0$ , which in the divided-word notation is $[P_{i (2) j}] - [P_{i j i}] + [P_{j i (2)}] = 0$ , i.e. $[P_{i j i}] = [P_{i (2) j}] + [P_{j i (2)}]$ .

Depends on

Used by

Degenerate parameters Qᵢⱼ destroy the faithful polynomial-representation route Counterexample

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups I, Representation Theory 13 (2009) 309-347, arXiv:0803.4121v2 (standard reference, not scraped)
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Transactions of the AMS 363 (2011) 2685-2700, arXiv:0804.2080v1 (standard reference, not scraped)
Raphael Rouquier, 2-Kac-Moody Algebras, arXiv:0812.5023v3 (66 pages) (standard reference, not scraped)