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The coinvariant algebra of S_3 as the Soergel bimodule of the longest element

Example

Take $n = 3$ and let $w_{0} = s_{1} s_{2} s_{1}$ be the longest element of $S_{3}$ . Then $B_{w_{0}} = S \otimes_{S^{S_{3}}} S (3)$ and its reduced module is $C (3)$ , where $C = S / (S_{+}^{S_{3}})$ is the unshifted coinvariant algebra: $C$ has the Poincare polynomial $1 + 2 t^{2} + 2 t^{4} + t^{6}$ , so $\dim C = 6$ , and in the centered grading $\dim (\overline{B_{w_{0}}})^{- 3} = \dim (\overline{B_{w_{0}}})^{3} = 1, \dim (\overline{B_{w_{0}}})^{- 1} = \dim (\overline{B_{w_{0}}})^{1} = 2.$ The standard-basis expansion of the Kazhdan-Lusztig basis element is ${\underline{H}}_{w_{0}} = H_{w_{0}} + v (H_{s_{1} s_{2}} + H_{s_{2} s_{1}}) + v^{2} (H_{s_{1}} + H_{s_{2}}) + v^{3},$ obtained by solving the bar-invariance and triangularity conditions with coefficients in $v^{d} Z [v^{- 2}]$ ; its coefficients are non-negative, its value at $v = 1$ sums to $6 = \dim C$ , and it agrees with the rank-one and rank-two Kazhdan-Lusztig data $P_{e, w_{0}} = P_{s_{i}, w_{0}} = P_{w, w_{0}} = 1$ for $ℓ (w) = 2$ . Hence $ch (B_{w_{0}}) = {\underline{H}}_{w_{0}}$ : Soergel's conjecture holds for the longest element of $S_{3}$ , and the $Δ$ -multiplicities of the coinvariant bimodule are the six coefficients just displayed, one for each element of $S_{3}$ .

Verification

technique · direct; the identification with the invariants-tensor bimodule, an explicit bar-invariance computation in the Hecke algebra, and the comparison of the graded dimensions

Given: $S = R [x_{1}, x_{2}, x_{3}]$ with $\deg x_{i} = 2$ , the longest element $w_{0} = s_{1} s_{2} s_{1}$ , the rank-two longest bimodule $B_{i, i + 1, i} = R \otimes_{R^{W_{i, i + 1}}} R (3)$ (The rank-two longest type-A Soergel bimodule), the reduced module and the character (The reduced Soergel module and the dominant Lefschetz operator, Standard graph bimodules, support filtrations and characters), the normalized Hecke algebra with $H_{s}^{2} = 1 + (v^{- 1} - v) H_{s}$ and the Kazhdan-Lusztig basis (Existence and uniqueness of the Kazhdan–Lusztig basis), and multiplicativity of the standard character (The type-A standard character is multiplicative).

[F1] The rank-two longest bimodule is $S \otimes_{S^{S_{3}}} S (3)$ after real scalar extension, and right reduction is tensoring with $S / S_{+} = R$ (The rank-two longest type-A Soergel bimodule, The reduced Soergel module and the dominant Lefschetz operator). The symmetric-polynomial and quotient-basis computations needed here are supplied in step 1.1; no general invariant-ring theorem is assumed.

[F2] In the Hecke algebra $H_{s}^{- 1} = H_{s} + (v - v^{- 1})$ for every simple $s$ , and the bar involution acts on the standard basis by $\overline{H_{w}} = H_{w^{- 1}}^{- 1}$ ; the Kazhdan-Lusztig basis is the unique basis of elements ${\underline{H}}_{w} = H_{w} + \sum_{y < w} p_{y, w} H_{y}$ with $p_{y, w} \in v Z [v]$ and $\overline{{\underline{H}}_{w}} = {\underline{H}}_{w}$ (Existence and uniqueness of the Kazhdan–Lusztig basis).

[F3] The coefficient of $H_{y}$ in $ch (M)$ is $\sum_{d} (M : Δ_{y} (d)) v^{d}$ with non-negative integral multiplicities, and for the standard layer $Δ_{y} (d)$ the reduced dimension sits in degree $ℓ (y) - d$ ; hence the multiset of monomials of the character, read with the degrees $ℓ (y) - d$ , is the graded dimension of the reduced module (Standard graph bimodules, support filtrations and characters).

[F4] The character is multiplicative with simple value $H_{s} + v$ , and the rank-two decomposition is $B_{s_{1}} B_{s_{2}} B_{s_{1}} ≅ B_{w_{0}} \oplus B_{s_{1}}$ (The type-A standard character is multiplicative, Rank-two type-A Soergel bimodule decompositions).

1.1givenF1

The bimodule and its reduction. Symmetric polynomials in three variables are generated by $e_{1}, e_{2}, e_{3}$ : for a homogeneous symmetric polynomial, its lexicographically leading monomial has exponents $a \geq b \geq c$ ; subtracting its coefficient times $e_{1}^{a - b} e_{2}^{b - c} e_{3}^{c}$ lowers that leading monomial. The finite set of monomials of the fixed total degree makes this reduction terminate. Thus $S^{S_{3}} = R [e_{1}, e_{2}, e_{3}]$ and [F1] gives the reduced module $C (3)$ with $C = S / (e_{1}, e_{2}, e_{3})$ . Eliminating $x_{3} = - x - y$ gives $C = R [x, y] / (x^{3}, y^{2} + x y + x^{2})$ : the $e_{3}$ relation is $x^{2} y + x y^{2} = 0$ , and subtracting this from $x (x^{2} + x y + y^{2}) = 0$ gives $x^{3} = 0$ , with the same identities proving the reverse inclusion of ideals. Over $R [x] / (x^{3})$ , division by the monic quadratic in $y$ gives the free basis $1, y$ ; hence $1, x, y, x^{2}, x y, x^{2} y$ is a real basis and the dimensions are $1, 2, 2, 1$ in degrees $0, 2, 4, 6$ , centered at $- 3, - 1, 1, 3$ after shifting by $(3)$ . The tensor bimodule is generated as a bimodule by $1 \otimes 1$ and its bottom component is one dimensional, so every degree-zero endomorphism is scalar; it is therefore indecomposable. The balanced surjection $f \otimes g \mapsto f w_{0} (g)$ onto $S_{w_{0}}$ proves that it has the $w_{0}$ graph in its support. In the rank-two decomposition it is the summand carrying the $w_{0}$ graph (the other summand $B_{s_{1}}$ has only the $e, s_{1}$ graphs), so its label is $w_{0}$ .

1.2givenF2algebra

The Kazhdan-Lusztig element. Put $A : = v - v^{- 1}$ and $J : = H_{w_{0}} + v (H_{s_{1} s_{2}} + H_{s_{2} s_{1}}) + v^{2} (H_{s_{1}} + H_{s_{2}}) + v^{3}$ . The element $J$ lies in $H_{w_{0}} + \sum_{y < w_{0}} v Z [v] H_{y}$ , since all coefficients belong to $v Z [v]$ . For bar-invariance we expand: from $H_{s}^{- 1} = H_{s} + A$ and the reduced products $H_{s_{1}} H_{s_{2}} = H_{s_{1} s_{2}}$ and $H_{s_{1} s_{2}} H_{s_{1}} = H_{w_{0}}$ one computes $\overline{H_{s_{1} s_{2}}} = H_{s_{1} s_{2}} + A (H_{s_{1}} + H_{s_{2}}) + A^{2}$ and $\overline{H_{w_{0}}} = H_{w_{0}} + A (H_{s_{1} s_{2}} + H_{s_{2} s_{1}}) + A^{2} (H_{s_{1}} + H_{s_{2}}) + A + A^{3}$ , while $\overline{H_{s_{1}}} = H_{s_{1}} + A$ and $\overline{H_{s_{2}}} = H_{s_{2}} + A$ . Applying the semilinear involution to $J$ gives $\overline{J} = \overline{H_{w_{0}}} + v^{- 1} (\overline{H_{s_{1} s_{2}}} + \overline{H_{s_{2} s_{1}}}) + v^{- 2} (\overline{H_{s_{1}}} + \overline{H_{s_{2}}}) + v^{- 3}$ ; collecting coefficients in the standard basis, the coefficient of $H_{s_{1} s_{2}} + H_{s_{2} s_{1}}$ becomes $A + v^{- 1} = v$ , the coefficient of $H_{s_{1}} + H_{s_{2}}$ becomes $A^{2} + 2 v^{- 1} A + v^{- 2} = v^{2}$ , and the coefficient of $H_{e}$ becomes $A + A^{3} + 2 v^{- 1} A^{2} + 2 v^{- 2} A + v^{- 3} = v^{3}$ . Hence $\overline{J} = J$ , and by the uniqueness clause of [F2] $J = {\underline{H}}_{w_{0}}$ .

2.1step 1.1step 1.2F3F4∎

Character and multiplicities. By [F4], $ch (B_{w_{0}}) = (H_{s_{1}} + v) (H_{s_{2}} + v) (H_{s_{1}} + v) - (H_{s_{1}} + v) = J$ , the last equality following by the quadratic relation $H_{s}^{2} = 1 + (v^{- 1} - v) H_{s}$ and reduced multiplication. Step 1.2 gives $J = {\underline{H}}_{w_{0}}$ , so the $Δ$ -multiplicities of $B_{w_{0}}$ are exactly the six coefficients of $J$ , one for each of the six elements of $S_{3}$ . Reading them through [F3], the layer $Δ_{w_{0}} (0)$ contributes in degree $ℓ (w_{0}) = 3$ , the two layers $Δ_{s_{1} s_{2}} (1), Δ_{s_{2} s_{1}} (1)$ contribute in degree $1$ with multiplicity two, the two layers $Δ_{s_{1}} (2), Δ_{s_{2}} (2)$ contribute in degree $- 1$ with multiplicity two and the layer $Δ_{e} (3)$ contributes in degree $- 3$ ; this is the graded dimension $1, 2, 2, 1$ of the centered coinvariant algebra computed in step 1.1, and the value at $v = 1$ is $6 = \dim C$ . The coefficients $P_{e, w_{0}} = P_{s_{i}, w_{0}} = P_{s_{1} s_{2}, w_{0}} = P_{s_{2} s_{1}, w_{0}} = 1$ are the classical normalization of these coefficients. This verifies the example.

Remarks

The two computations are independent: the bar-invariance determines the Kazhdan-Lusztig element from the Hecke algebra alone, and the graded dimensions of the coinvariant algebra come from the invariant theory; their agreement is the content of Soergel's conjecture at $w_{0}$ .
The example is an illustration of the character theorem, not a supplier of it; the base of the simultaneous induction is computed on the axiom page and does not use this page.

Depends on

Used by

Hard Lefschetz and the Hodge-Riemann signs for the coinvariant algebra of S₃ Example

Dependency tree · two levels

33 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

Elias-Williamson, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2, final version; Ann. of Math. 180 (2014) 1089-1136) (standard reference, not scraped)
Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ueber Polynomringen (arXiv:math/0403496, published J. Inst. Math. Jussieu 6 (2007) 501-525) (standard reference, not scraped)
Riche, La theorie de Hodge des bimodules de Soergel (d'apres Soergel et Elias-Williamson), Seminaire Bourbaki no. 1139 (arXiv:1711.02464, Octobre 2017) (standard reference, not scraped)