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The dual trace basis for the type-A2 Hecke algebra

Example

Take the equal-parameter Hecke algebra of $W = S_{3}$ over $R = Z [v^{\pm 1}]$ , with basis

1, T_{1}, T_{2}, T_{12}, T_{21}, T_{w_{0}}, T_{12} = T_{1} T_{2}, T_{21} = T_{2} T_{1}, T_{w_{0}} = T_{1} T_{2} T_{1} = T_{2} T_{1} T_{2} .

For the canonical trace $τ (T_{w}) = δ_{w, 1}$ , the inverse map on $W$ is

1^{- 1} = 1, s_{1}^{- 1} = s_{1}, s_{2}^{- 1} = s_{2}, (s_{1} s_{2})^{- 1} = s_{2} s_{1}, (s_{2} s_{1})^{- 1} = s_{1} s_{2}, w_{0}^{- 1} = w_{0} .

Thus the Gram matrix $(τ (T_{x} T_{y}))_{x, y \in W}$ in the displayed ordered basis is

(\begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array})

The trace-dual basis is $(1, T_{1}, T_{2}, T_{21}, T_{12}, T_{w_{0}})$ , so the dual vector to $T_{w}$ is $T_{w^{- 1}}$ . It is not generally the algebra inverse: if $a = v - v^{- 1}$ , then $T_{1}^{- 1} = T_{1} - a$ , whereas the trace-dual vector to $T_{1}$ is $T_{1}$ .

Facts & Assumptions

Given: The equal-parameter type-A2 Hecke algebra over $R = Z [v^{\pm 1}]$ , its standard basis, and the coefficient-at-identity trace.

[F1]

For finite $W$ , the canonical trace sends $\sum_{w} r_{w} T_{w}$ to $r_{1}$ . (The canonical Hecke trace and symmetrizing form)

[F2]

For all $x, y \in W$ , $τ (T_{x} T_{y}) = δ_{x^{- 1}, y}$ , and the trace-dual basis is ${T_{w^{- 1}} : w \in W}$ . (The canonical Hecke trace is symmetric and perfect)

[F3]

$R = Z [v^{\pm 1}]$ is the Laurent polynomial ring and $T_{i}^{2} = (v - v^{- 1}) T_{i} + 1$ . (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy)

[F4]

The standard elements $T_{w}$ form an $R$ -basis, with multiplication $T_{x} T_{y} = T_{x y}$ for additive lengths. (The standard basis of the generic Hecke algebra and base change, Reduced-word independence of T_w and the length-multiplication rules)

Verification

Given: The data in Facts & Assumptions.

1.1F4given

The reduced words for the six elements of $S_{3}$ give exactly the listed standard basis. Reversing each reduced word gives the displayed inverse map; in particular $w_{0} = s_{1} s_{2} s_{1}$ is its own inverse.

2.1F1F2step 1.1

By [F1], $τ$ is the coefficient-at-identity functional; applying [F2] to the ordered basis in step 1.1 gives $τ (T_{x} T_{y}) = 1$ exactly when the column index is $y = x^{- 1}$ , and $0$ otherwise. The resulting matrix is the displayed permutation matrix.

3.1F2step 2.1algebra

The matrix in step 2.1 permutes the coordinate basis by $x \mapsto x^{- 1}$ , so it has an inverse equal to itself and determinant $- 1$ , a unit in $R$ . Its dual vectors are therefore $T_{x^{- 1}}$ ; directly, $τ (T_{x} T_{y^{- 1}}) = δ_{x, y}$ .

4.1F3step 3.1algebra∎

The quadratic relation in [F3] gives $T_{1}^{2} = a T_{1} + 1$ , so $T_{1} (T_{1} - a) = 1 = (T_{1} - a) T_{1}$ and $T_{1}^{- 1} = T_{1} - a$ . The Laurent ring $R$ has distinct basis monomials $v$ and $v^{- 1}$ , hence $a = v - v^{- 1} \neq 0$ ; thus $T_{1}^{- 1} \neq T_{1}$ , while step 3.1 shows $T_{1}$ itself is trace-dual to $T_{1}$ . This distinguishes the two notions explicitly.

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Sources

Andrew Mathas, Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group (self-contained notes, 1998; complete 79-page text) (standard reference, not scraped)
Meinolf Geck, Modular Representations of Hecke Algebras (EPFL lecture notes, arXiv:math/0511548v2) (standard reference, not scraped)