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edited Mar 20, 2021 at 11:53

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Positivity of Iwahori.HeckeIwahori–Hecke algebra characters on the Kazhdan-Lusztig basis

I'm$\DeclareMathOperator\tr{tr}$I'm interested in the irreducible characters of a finite Iwahori-HeckeIwahori–Hecke algebra evaluated at the Kazhdan-LusztigKazhdan–Lusztig basis. These are Laurent polynomials.

Are the coefficients of these polynomials always positive?

In type $A_n$, the irreducible representations are given by left cells. Denote by $(C_w)$ the Kazhdan-Lusztig basis and by $(C^w)$ its dual basis with respect to the standard trace $tr$$\tr$. Take $\Lambda$ such a left cell and denote by $\chi_\lambda$ the associated character. It is known that $\chi_\lambda(h) = tr(\sum_{x \in \Lambda}C_xC^xh)$$\chi_\lambda(h) = \tr(\sum_{x \in \Lambda}C_xC^xh)$. Hence $$\chi_\lambda(C_w) = \sum_{x\in\Lambda} tr(C_xC^xC_w)$$$$\chi_\lambda(C_w) = \sum_{x\in\Lambda} \tr(C_xC^xC_w)$$ which is positive since the expression equals a sum of structure constants in the KL-basis.

In the literature, one finds character tables, but always on the standard basis, not the KL-basis. Is there a reference where the character tables on the KL basis are computed?

Finally, most of the positivity properties in the Hecke algebras can be proven using the categorification with Soergel bimodules. Is there a categorification of the irreducible characters?

Positivity of Iwahori.Hecke algebra characters on the Kazhdan-Lusztig basis

I'm interested in the irreducible characters of a finite Iwahori-Hecke algebra evaluated at the Kazhdan-Lusztig basis. These are Laurent polynomials.

Are the coefficients of these polynomials always positive?

In type $A_n$, the irreducible representations are given by left cells. Denote by $(C_w)$ the Kazhdan-Lusztig basis and by $(C^w)$ its dual basis with respect to the standard trace $tr$. Take $\Lambda$ such a left cell and denote by $\chi_\lambda$ the associated character. It is known that $\chi_\lambda(h) = tr(\sum_{x \in \Lambda}C_xC^xh)$. Hence $$\chi_\lambda(C_w) = \sum_{x\in\Lambda} tr(C_xC^xC_w)$$ which is positive since the expression equals a sum of structure constants in the KL-basis.

In the literature, one finds character tables, but always on the standard basis, not the KL-basis. Is there a reference where the character tables on the KL basis are computed?

Finally, most of the positivity properties in the Hecke algebras can be proven using the categorification with Soergel bimodules. Is there a categorification of the irreducible characters?

Positivity of Iwahori–Hecke algebra characters on the Kazhdan-Lusztig basis

$\DeclareMathOperator\tr{tr}$I'm interested in the irreducible characters of a finite Iwahori–Hecke algebra evaluated at the Kazhdan–Lusztig basis. These are Laurent polynomials.

Are the coefficients of these polynomials always positive?

In type $A_n$, the irreducible representations are given by left cells. Denote by $(C_w)$ the Kazhdan-Lusztig basis and by $(C^w)$ its dual basis with respect to the standard trace $\tr$. Take $\Lambda$ such a left cell and denote by $\chi_\lambda$ the associated character. It is known that $\chi_\lambda(h) = \tr(\sum_{x \in \Lambda}C_xC^xh)$. Hence $$\chi_\lambda(C_w) = \sum_{x\in\Lambda} \tr(C_xC^xC_w)$$ which is positive since the expression equals a sum of structure constants in the KL-basis.

In the literature, one finds character tables, but always on the standard basis, not the KL-basis. Is there a reference where the character tables on the KL basis are computed?

Finally, most of the positivity properties in the Hecke algebras can be proven using the categorification with Soergel bimodules. Is there a categorification of the irreducible characters?

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asked Mar 17, 2021 at 18:00

AThomas

Positivity of Iwahori.Hecke algebra characters on the Kazhdan-Lusztig basis

I'm interested in the irreducible characters of a finite Iwahori-Hecke algebra evaluated at the Kazhdan-Lusztig basis. These are Laurent polynomials.

Are the coefficients of these polynomials always positive?

In type $A_n$, the irreducible representations are given by left cells. Denote by $(C_w)$ the Kazhdan-Lusztig basis and by $(C^w)$ its dual basis with respect to the standard trace $tr$. Take $\Lambda$ such a left cell and denote by $\chi_\lambda$ the associated character. It is known that $\chi_\lambda(h) = tr(\sum_{x \in \Lambda}C_xC^xh)$. Hence $$\chi_\lambda(C_w) = \sum_{x\in\Lambda} tr(C_xC^xC_w)$$ which is positive since the expression equals a sum of structure constants in the KL-basis.

In the literature, one finds character tables, but always on the standard basis, not the KL-basis. Is there a reference where the character tables on the KL basis are computed?

Finally, most of the positivity properties in the Hecke algebras can be proven using the categorification with Soergel bimodules. Is there a categorification of the irreducible characters?