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The type-A standard character is multiplicative

Statement

Let $n \geq 2$ and $k = Q$ , let $K_{0}^{s p l i t} ({S B i m}_{n})$ be the split Grothendieck ring of the type-A Soergel category (Split Grothendieck rings of the type-A Soergel categories) and let $h_{Δ}, h_{\nabla}$ be the intrinsic $Δ$ - and $\nabla$ -multiplicity characters of Standard graph bimodules, support filtrations and characters, which by The type-A support filtration multiplicities are intrinsic are well-defined functions of the bimodule and additive on direct sums, so that they descend to $Z$ -linear maps on $K_{0}^{s p l i t} ({S B i m}_{n})$ with values in the Hecke algebra $H_{S_{n}}$ of The type-A Hecke algebra in Soergel normalization whose normalized generators are $H_{i} = v (T_{i} + 1)$ . Then:

$h_{Δ}$ is a homomorphism of $Z [v, v^{- 1}]$ -algebras $h_{Δ} : K_{0}^{s p l i t} ({S B i m}_{n}) ⟶ H_{S_{n}}, h_{Δ} (X Y) = h_{Δ} (X) h_{Δ} (Y), h_{Δ} (v X) = v h_{Δ} (X),$ that is $h_{Δ} (v X) = v h_{Δ} (X)$ , and $h_{Δ} ([B_{i}]) = H_{i}$ for every simple reflection;
$h_{\nabla}$ is multiplicative, $h_{\nabla} (X Y) = h_{\nabla} (X) h_{\nabla} (Y)$ , and semilinear for the involution $v \mapsto v^{- 1}$ of the coefficients: $h_{\nabla} (v X) = v^{- 1} h_{\nabla} (X)$ and $h_{\nabla} ([B_{i}]) = H_{i}$ ; equivalently $h_{\nabla} = d \circ h_{Δ}$ , where $d$ is the semilinear algebra involution of $H_{S_{n}}$ determined by $d (v) = v^{- 1}$ and $d (T_{i}) = T_{i}^{- 1}$ ;
for every Bott–Samelson word $\underline{i} = (i_{1}, \dots, i_{r})$ one has $h_{Δ} (B_{\underline{i}}) = h_{\nabla} (B_{\underline{i}}) = H_{i_{1}} \dots H_{i_{r}}$ , and for every bimodule $M$ in $F_{Δ} \cap F_{\nabla}$ and simple reflection $s_{i}$ the simple-generator recursion $h_{Δ} (B_{i} \otimes_{R} M) = H_{i} h_{Δ} (M)$ , $h_{\nabla} (B_{i} \otimes_{R} M) = H_{i} h_{\nabla} (M)$ holds; in the Elias–Williamson grading shift $(1) = {- 1}$ this says that $(1)$ multiplies $h_{Δ}$ by $v$ and $h_{\nabla}$ by $v^{- 1}$ .

Facts & Assumptions

Given: The type-A Soergel category ${S B i m}_{n}$ with its split Grothendieck ring, the intrinsic characters $h_{Δ}, h_{\nabla}$ , the Hecke algebra $H_{S_{n}}$ with generators $T_{i}$ and normalized generators $H_{i} = v (T_{i} + 1)$ and $q = v^{- 2}$ , and the categorification isomorphism $Φ : K_{0}^{s p l i t} ({S B i m}_{n}) \to H_{S_{n}}$ .

[F1]

$Δ_{x} (d) = R_{x} {ℓ (x) - d}$ , $\nabla_{x} (d) = R_{x} {- ℓ (x) - d}$ , and $h_{Δ} (M) = \sum (M : Δ_{x} (d)) v^{d} {\tilde{T}}_{x}$ , $h_{\nabla} (M) = \sum (M : \nabla_{x} (d)) v^{- d} {\tilde{T}}_{x}$ , with ${\tilde{T}}_{x} = v^{ℓ (x)} T_{x}$ (Standard graph bimodules, support filtrations and characters).

[F2]

The graded multiplicities $(M : Δ_{x} (d))$ and $(M : \nabla_{x} (d))$ are independent of the flag enumeration, and they are additive over direct sums and over direct summands lying in $F_{Δ} \cap F_{\nabla}$ ; hence $h_{Δ}$ and $h_{\nabla}$ are intrinsic and additive, and descend to $Z$ -linear maps on the split Grothendieck group (The type-A support filtration multiplicities are intrinsic).

[F3]

For every simple reflection $s = s_{i}$ and every $M \in F_{Δ} \cap F_{\nabla}$ : both $B_{i} \otimes_{R} M$ and $M \otimes_{R} B_{i}$ lie in $F_{Δ} \cap F_{\nabla}$ ; $h_{Δ} (B_{i} \otimes_{R} M) = H_{i} h_{Δ} (M)$ and $h_{\nabla} (B_{i} \otimes_{R} M) = H_{i} h_{\nabla} (M)$ ; $h_{Δ} (M {k}) = v^{- k} h_{Δ} (M)$ and $h_{\nabla} (M {k}) = v^{k} h_{\nabla} (M)$ ; and $h_{Δ} (B_{i}) = h_{\nabla} (B_{i}) = H_{i}$ , $h_{Δ} (R) = h_{\nabla} (R) = 1$ (The type-A character recursion under simple Soergel tensoring).

[F4]

There is an isomorphism of $Z [v, v^{- 1}]$ -algebras $Φ : K_{0}^{s p l i t} ({S B i m}_{n}) \to H_{S_{n}}$ with $Φ ([B_{i}]) = H_{i}$ , $Φ (v X) = v Φ (X)$ and $Φ^{- 1} (T_{i}) = v^{- 1} [B_{i}] - 1$ ; it is the unique algebra isomorphism with $Φ ([B_{i}]) = H_{i}$ and $Φ (v) = v$ , because the classes of the $B_{i}$ together with $v^{\pm 1}$ generate $K_{0}^{s p l i t} ({S B i m}_{n})$ (The split Grothendieck group of the Soergel category is the type-A Hecke algebra).

[F5]

The Hecke algebra $H_{n}$ is presented by the $T_{i}$ with $T_{i}^{2} = (q - 1) T_{i} + q$ , the braid relations and the distant commutations, $q = v^{- 2}$ is a unit of $A = Z [v, v^{- 1}]$ , $H_{i} = v (T_{i} + 1)$ satisfies $H_{i}^{2} = (v + v^{- 1}) H_{i}$ and $T_{i} = v^{- 1} H_{i} - 1$ ; the products of the $H_{i}$ along reduced words form a triangular $A$ -basis with unit diagonal against the normalized standard basis ${{\tilde{T}}_{w} = v^{ℓ (w)} T_{w}}$ , which is itself an $A$ -basis (The type-A Hecke algebra in Soergel normalization, The standard basis of the type-A Hecke algebra and its multiplication rule).

[F6]

Every Soergel object is a graded summand of a finite sum of shifted words (The type-A Soergel category ${S B i m}_{n}$ ). Words lie in both flag categories (Bott–Samelson bimodules carry delta and nabla support filtrations), and each flag category is closed under sums, shifts and summands (Remark (d)(2) of Standard graph bimodules, support filtrations and characters, Soergel Bemerkung 5.5 and its dual). Hence every Soergel object belongs to $F_{Δ} \cap F_{\nabla}$ .

Proof

1.1

Descent to the split Grothendieck group: [F6] first puts every object of ${S B i m}_{n}$ in $F_{Δ} \cap F_{\nabla}$ . Thus [F2] applies to every such object: the multiplicities $(M : Δ_{x} (d))$ and $(M : \nabla_{x} (d))$ depend only on $M$ and are additive on direct sums, so the sums of [F1] are well defined on isomorphism classes and satisfy $h_{∙} (X \oplus Y) = h_{∙} (X) + h_{∙} (Y)$ ; consequently $h_{Δ}$ and $h_{\nabla}$ descend to $Z$ -linear maps on $K_{0}^{s p l i t} ({S B i m}_{n})$ . By the shift clause of [F3], $h_{Δ} (v X) = h_{Δ} (X {- 1}) = v h_{Δ} (X)$ and $h_{\nabla} (v X) = h_{\nabla} (X {- 1}) = v^{- 1} h_{\nabla} (X)$ .

F1F2F3F6

1.2

Values on Bott–Samelson words: iterating the recursion of [F3] with $M = R$ gives $h_{Δ} (B_{i_{1}} \otimes_{R} \dots \otimes_{R} B_{i_{r}}) = H_{i_{1}} \dots H_{i_{r}} h_{Δ} (R) = H_{i_{1}} \dots H_{i_{r}}$ and likewise $h_{\nabla} (B_{i_{1}} \otimes_{R} \dots \otimes_{R} B_{i_{r}}) = H_{i_{1}} \dots H_{i_{r}}$ ; for $r = 0$ this is $h_{Δ} (R) = h_{\nabla} (R) = 1$ , and for $r = 1$ it gives $h_{Δ} ([B_{i}]) = h_{\nabla} ([B_{i}]) = H_{i}$ .

1.3

The semilinear involution $d$ : since $q$ is a unit, the quadratic relation $T_{i}^{2} = (q - 1) T_{i} + q$ is equivalent to $T_{i}^{- 1} = q^{- 1} T_{i} + q^{- 1} - 1$ . Thus assigning $d (v) = v^{- 1}$ and $d (T_{i}) = T_{i}^{- 1}$ respects the quadratic relation with $q$ replaced by $q^{- 1}$ . It respects distant commutations because commuting elements have commuting inverses, and it respects the rank-two braid relation because $(T_{i} T_{i + 1} T_{i})^{- 1} = T_{i}^{- 1} T_{i + 1}^{- 1} T_{i}^{- 1}$ and likewise for its other side. Hence the assignment extends to a semilinear algebra endomorphism of $H_{n}$ ; applying it twice fixes $v$ and every $T_{i}$ , so $d^{2} = i d$ . Directly $d (H_{i}) = d (v (T_{i} + 1)) = v^{- 1} (T_{i}^{- 1} + 1) = v^{- 1} (q^{- 1} T_{i} + q^{- 1}) = v T_{i} + v = H_{i}$ , using $v^{- 1} q^{- 1} = v$ because $q = v^{- 2}$ .

2.1

Identification of the $Δ$ -character: the maps $h_{Δ}$ and $Φ$ of [F4] are both additive and $Z [v, v^{- 1}]$ -linear by step 1.1 and by [F4], and they agree on every product of the generators $[B_{i}]$ and $v^{\pm 1}$ : on a product of the $[B_{i}]$ along a word this is step 1.2 together with the multiplicativity of $Φ$ , and $v$ acts by multiplication by $v$ on both sides; since the classes of the $B_{i}$ together with $v^{\pm 1}$ generate $K_{0}^{s p l i t} ({S B i m}_{n})$ as an algebra by [F4], the two maps agree everywhere on the additive span of the monomials, hence $h_{Δ} = Φ$ ; therefore $h_{Δ}$ is multiplicative and $Z [v, v^{- 1}]$ -linear, with $h_{Δ} ([B_{i}]) = H_{i}$ .

F4step 1.1step 1.2

2.2

Identification of the $\nabla$ -character: by step 1.3 the map $d$ is a semilinear algebra involution with $d (H_{i}) = H_{i}$ , so $d \circ h_{\nabla}$ is $Z [v, v^{- 1}]$ -linear by the semilinearity of $d$ and of $h_{\nabla}$ in step 1.1, and it agrees with $Φ$ on the word products by step 1.2: $(d \circ h_{\nabla}) (B_{\underline{i}}) = d (H_{i_{1}} \dots H_{i_{r}}) = H_{i_{1}} \dots H_{i_{r}} = Φ ([B_{\underline{i}}])$ ; hence $d \circ h_{\nabla} = Φ$ on the generating monomials, so $d \circ h_{\nabla} = Φ$ on all of $K_{0}^{s p l i t} ({S B i m}_{n})$ by [F4], and applying the involution $d$ gives $h_{\nabla} = d \circ Φ$ . Since $d$ is an algebra endomorphism and $Φ$ an algebra isomorphism, $h_{\nabla} (X Y) = h_{\nabla} (X) h_{\nabla} (Y)$ , and $h_{\nabla} (v X) = v^{- 1} h_{\nabla} (X)$ by step 1.1.

F4step 1.1step 1.2step 1.3

3.1

Recursion and conclusion: claim (3) is the recursion and shift clauses of [F3] together with step 1.2, read in the Elias–Williamson shift $(1) = M {- 1}$ ; claims (1) and (2) are steps 2.1 and 2.2, including the displayed values $h_{Δ} ([B_{i}]) = h_{\nabla} ([B_{i}]) = H_{i}$ and the identification $h_{\nabla} = d \circ h_{Δ}$ ; the multiplicativity thus proved is a statement about the whole split Grothendieck ring, obtained from the categorification isomorphism rather than from tensoring two graph layers alone. ∎

F3F4step 1.2step 2.1step 2.2

Remark

(a) What is imported and what is computed here. The recursion of [F3] is Soergel's Propositions 5.7 and 5.9 as recorded and proved in The type-A character recursion under simple Soergel tensoring; it alone does not give multiplicativity on the whole ring, because the Hecke product of two arbitrary classes is not computable from two flag layers. The identification of $h_{Δ}$ with the categorification isomorphism $Φ$ of The split Grothendieck group of the Soergel category is the type-A Hecke algebra is what upgrades the generator recursion to a ring homomorphism, and the semilinear Hecke involution $d$ transports that to $h_{\nabla}$ .

(b) Why the involution fixes $H_{i}$ . The two generators are related by $H_{i} = v (T_{i} + 1)$ ; since $q = v^{- 2}$ , the substitution $T_{i} \mapsto T_{i}^{- 1}$ and $v \mapsto v^{- 1}$ fixes both summands of $v (T_{i} + 1)$ , as the computation of step 1.3 shows. This is the semilinear involution used by Soergel's duality argument; no Kazhdan–Lusztig positivity statement and no conjecture of the sources is used.

(c) Choice. No choice principle is used: the characters are computed from the fixed flag structures of the objects, the word classes are finite products of generators, and the identification of the two maps is a comparison on generating monomials.

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Sources

Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, §§5–6 (standard reference, not scraped)
Elias–Williamson, Soergel Calculus, §§3, 5–7 (standard reference, not scraped)