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The Hecke bar involution is well defined
Facts & Assumptions
Given: The presented algebra , its generators , the coefficient involution , and the generator assignment from The normalized type-A Hecke algebra and its bar involution.
The defining relations are , the adjacent braid relations, and the distant commutations; the quadratic relation gives (The normalized type-A Hecke algebra and its bar involution).
On the free algebra, the candidate assignment is and ; in the quotient the quadratic relation identifies this latter element with (The normalized type-A Hecke algebra and its bar involution).
Each is the product of the generators along a reduced expression for , and the elements form the standard basis (The normalized type-A Hecke algebra and its bar involution, The standard basis of the generic type-A Hecke algebra).
Statement
In there is a unique -semilinear unital ring involution, denoted by a bar, with and . It is multiplicative and satisfies and for every ; consequently each is invertible and .
Proof
Uniqueness and inverse generators. Any semilinear ring homomorphism with the prescribed coefficient action and generator images is unique: its action on is fixed, and the generate as an -algebra. Put . By the quadratic relation, , so . Multiplying the quadratic relation by gives , the quadratic relation with replaced by .
The assignment respects the presentation. On the free associative algebra, extend and semilinearly and multiplicatively as in [F2]. In the quotient, step 1.1 identifies with . The inverse quadratic relation in step 1.1 shows that the image of each quadratic relator is zero in the quotient. The braid relator maps to the equality obtained by inverting both sides of ; the words are palindromes. A distant commutation relator maps to the commutation of the inverse generators, which follows by inverting the original equality. Thus the defining ideal is preserved and the assignment descends to a unital semilinear algebra endomorphism of .
Involutivity. Applying bar twice fixes . Since a ring homomorphism sends the inverse of a unit to the inverse of its image, . It therefore fixes every generator and coefficient, so bar squared is the identity.
Formula on the standard basis. Let be reduced. By multiplicativity, . This includes , for which the product is empty. Every generator is a unit by step 1.1, hence every is a unit, and applying bar gives the equivalent formula . This proves the statement. The case has no generators and reduces to the coefficient involution of .
Depends on
Used by
- Bruhat intervals and the R-coefficients Definition
- The Kazhdan–Lusztig bases of S₂ and S₃ Example
- Reversal anti-involution commutes with the Hecke bar Lemma
- Existence and uniqueness of the Kazhdan–Lusztig basis Theorem
- Kazhdan–Lusztig cells of type A are classified by RSK tableaux Theorem
- The R-coefficient recursion, support, degree bounds and inversion Theorem
Dependency tree · two levels
5 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
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 (45 pp.) — §3.2 (printed pp. 15–16): the Hecke algebra in the normalization $H_xH_s=H_{xs}$ or $(v^{-1}-v)H_x+H_{xs}$, the bar involution $\overline{H_x}=H_{x^{-1}}^{-1}$, the Kazhdan–Lusztig basis $\{\underline H_x\}$ characterized by bar-invariance and $\underline H_x\in H_x+\sum_{y<x}v\mathbb Z[v]H_y$, the example $\underline H_s=H_s+vH_{\mathrm{id}}$, and Remark 3.2: $v=q^{-1/2}$, $H_x=v^{\ell(x)}T_x$, $\underline H_x=C'_x$, $h_{y,x}=v^{\ell(x)-\ell(y)}P_{y,x}(v^{-2})$ (standard reference, not scraped)
- G. Lusztig, Hecke Algebras with Unequal Parameters (revised book version, arXiv:math/0208154v2) — the split case $L\equiv1$ read as the source for the bar operator, the R-coefficients, the new basis, its multiplication properties and cells; translated to the normalization of this page by $v_L=v^{-1}$ (so $v_L^{L(w)-L(y)}=v^{-(\ell(w)-\ell(y))}$) (standard reference, not scraped)