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Soergel's character conjecture for type A

Facts & Assumptions

Given: A symmetric group $S_{n}$ , the real type- $A$ Soergel category of The real type-A Soergel category by scalar extension with its normalized indecomposables $B_{w}^{R}$ and their characters, the type- $A$ Hecke algebra with its standard basis $H_{w}$ and Kazhdan-Lusztig basis ${\underline{H}}_{w}$ (The type-A Kazhdan–Lusztig basis has elementary triangular bar properties), the statement $S (w) : ch (B_{w}^{R}) = {\underline{H}}_{w}$ and the hypotheses $h L (w)$ , $H R (w)$ of the dictionary (Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR), and the deformed operator $L_{ζ}$ on $B_{x}^{R} B_{s}^{R}$ (The deformed Lefschetz operator L_zeta on B_x B_s).

[F1]

The simultaneous induction theorem: if a finite Bruhat ideal $X$ satisfies the parameterized ascent and positive-descent HR conditions, then $S (x)$ , $h L (x)$ and $H R (x)$ hold for every $x \in X$ ; the conditions hold for $X = S_{n}$ , so all three statements hold for every $w \in S_{n}$ , and for every $x, s$ with $x s > x$ the strong forms $H R b a r (x, s)_{ζ}$ hold for all $ζ \geq 0$ (Simultaneous induction: Soergel's conjecture, hard Lefschetz and the Hodge-Riemann relations hold for all elements).

[F2]

The standard character is the split-Grothendieck isomorphism $Φ : K_{0}^{s p l i t} \to H$ , not its inverse. In the normalized standard basis $H_{w} = v^{ℓ (w)} T_{w}$ , its simple value is $Φ ([B_{s}]) = v (T_{s} + 1) = H_{s} + v = {\underline{H}}_{s}$ . The published split-Grothendieck supplier names this simple Kazhdan--Lusztig generator $H_{i}$ ; that $H_{i}$ is distinct from the standard $H_{s}$ used here. The character is additive and multiplicative, satisfies $ch (M (d)) = v^{d} ch (M)$ , and on an object means $ch (M) = Φ ([M])$ . Real scalar extension preserves labels, shifts and support-filtration characters, and character identities over $R$ descend to $Q$ (The split Grothendieck group of the Soergel category is the type-A Hecke algebra, The type-A standard character is multiplicative, Indecomposable labels, characters and filtration ranks agree after real scalar extension).

[F3]

Descent: if $x s < x$ and $S (x)$ holds, then $B_{x} B_{s} = B_{x} (1) \oplus B_{x} (- 1)$ and the reduced operator is the matrix $(\begin{array}{cc} ρ \cdot (-) & 0 \\ ζ ρ (α_{s}^{\lor}) & ρ \cdot (-) \end{array})$ ; if in addition $h L (x)$ holds and $ζ > 0$ then $h L (x, s)_{ζ}$ holds, while centered hard Lefschetz fails at $ζ = 0$ (Right-descent Soergel bimodules descend to simple invariants under the character hypothesis, Hard Lefschetz for the deformed operator on B_x B_s, in the three cases).

Statement

For every $w \in S_{n}$ the character of the indecomposable type- $A$ Soergel bimodule equals the Kazhdan-Lusztig basis element: $ch (B_{w}^{R}) = {\underline{H}}_{w}$ . Equivalently, the class $[B_{w}^{R}]$ in the split Grothendieck group corresponds to ${\underline{H}}_{w}$ under the Hecke algebra isomorphism, and the extension of scalars of the rational $B_{w}$ categorifies the Kazhdan-Lusztig basis. Moreover the hard Lefschetz property $h L (w)$ and the Hodge-Riemann relations $H R (w)$ hold for every $w$ , with the standard signs, and the same identities hold for the rational category by the base-change comparison. Finally, for every right descent $x s < x$ the matrix of $L_{ζ}$ on the right reduction $\overline{B_{x} B_{s}} = \overline{B_{x}} (1) \oplus \overline{B_{x}} (- 1)$ is $(\begin{array}{cc} ρ \cdot (-) & 0 \\ ζ ρ (α_{s}^{\lor}) & ρ \cdot (-) \end{array})$ ; for $ζ > 0$ , $h L (x)$ implies $h L (x, s)_{ζ}$ without any additional character hypothesis, since $S$ is now established for all elements, while at $ζ = 0$ centered hard Lefschetz fails.

Proof

technique · direct; specialize the simultaneous induction to the full symmetric group and read off the character, the Hodge-theoretic statements and the unconditional descent consequence

1.1givenF1

Soergel's conjecture for all elements. The simultaneous induction theorem [F1] applies with $X = S_{n}$ : its two parameterized HR conditions are verified for every Bruhat ideal along the induction, and its conclusion gives $S (w)$ , $h L (w)$ and $H R (w)$ for every $w \in S_{n}$ . By the definition of $S (w)$ this is $ch (B_{w}^{R}) = {\underline{H}}_{w}$ for every $w$ , which is the character statement; the same theorem also gives the strong embedded forms $H R b a r (x, s)_{ζ}$ for all $x, s$ with $x s > x$ and all $ζ \geq 0$ .

2.1step 1.1F2

The class and the rational form. By [F2], $ch (M) = Φ ([M])$ for the split-Grothendieck isomorphism $Φ : K_{0}^{s p l i t} \to H$ , with $Φ ([B_{s}]) = H_{s} + v$ in the normalized standard basis. Hence $ch (B_{w}^{R}) = {\underline{H}}_{w}$ from step 1.1 means precisely $Φ ([B_{w}^{R}]) = {\underline{H}}_{w}$ . Real scalar extension preserves indecomposable labels, shifts and support-filtration multiplicities, so its character agrees with the rational character in the same integral Hecke algebra. The equality proved over $R$ therefore gives the same character equality over $Q$ , and the rational $B_{w}$ categorifies the same Kazhdan--Lusztig basis element. No inverse map from $H$ to $K_{0}$ is used as an object character.

2.2step 1.1F1

Hodge-theoretic statements. The hard Lefschetz property $h L (w)$ and the Hodge-Riemann relations $H R (w)$ for all $w$ are part of the conclusion of the simultaneous induction in step 1.1; the signs are the standard signs of the dictionary, normalized by the positive bottom Lefschetz form.

3.1step 1.1step 2.2F3

The unconditional descent consequence. Let $x s < x$ be a right descent. Since $S (x)$ now holds for every $x$ , the descent lemma [F3] displays the matrix of $L_{ζ}$ on $\overline{B_{x} B_{s}} = \overline{B_{x}} (1) \oplus \overline{B_{x}} (- 1)$ ; the positive-degree right-multiplication terms in the unreduced matrix vanish on precisely this reduction; combining it with $h L (x)$ from step 2.2 and $ζ > 0$ gives $h L (x, s)_{ζ}$ , with no character hypothesis beyond the already established $S (x)$ , and the same lemma records the failure of centered hard Lefschetz at $ζ = 0$ through the vanishing $(ℓ (x) + 1)$ -st power of $ρ$ on the bottom class of $B_{x} (1)$ while the opposite total degree is nonzero by self-duality.

4.1step 1.1step 2.1step 2.2step 3.1∎

Conclusion. Steps 1.1 and 2.1 give the character statement in the real and rational categories, step 2.2 the Hodge-theoretic statements with the standard signs, and step 3.1 the unconditional descent consequence; no step assumes positivity of Kazhdan-Lusztig polynomials, which is instead a corollary of the character identity, and no step uses a result beyond the suppliers listed. This proves the theorem.

Remarks

The theorem replaces the citation of Soergel's conjecture recorded on the Kazhdan-Lusztig basis page by a local proof: the character identity is a consequence of the simultaneous induction of this page, and positivity of the Kazhdan-Lusztig polynomials is deduced from it in the next item rather than assumed here.
The descent consequence is genuinely downstream: the descent lemma is applied only after $S (x)$ has been established for the element in question, so no logical cycle is created inside the induction.

Depends on

Used by

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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)