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Given a Weyl group $W$, Lusztig's $a$-function a from W to the natural numbers is defined as a function constant on two-sided cells and $a(w)=l(w)-2 d(w)$ when $w$ is an involution, where $l(w)$ is the length function and $d(w)$ is the degree of the Kazhdan-Lusztig polynomial $P_{1,w}$. Since every two-sided cell contains an involution the function a is defined on all of W. See for example before conjecture 15 in https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf for this defintion.

My question is how to obtain the $a$-function for the symmetric group as a Weyl group via SAGE.

A more explicit formula in this case is given as follows: Let $x$ be a permutation corresponding to the pair of tableau $(P(x),Q(x))$ by the Robinson-Schendsted correspondence and $shape(Q(x)')=( \lambda_1,...,\lambda_k)$ where $Q(x)'$ is the transposed tableaux. Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$, see for example exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups".

Given a Weyl group $W$, Lusztig's $a$-function a from W to the natural numbers is defined as a function constant on two-sided cells and $a(w)=l(w)-2 d(w)$ when $w$ is an involution, where $l(w)$ is the length function and $d(w)$ is the degree of the Kazhdan-Lusztig polynomial $P_{1,w}$. Since every two-sided cell contains an involution the function a is defined on all of W. See for example before conjecture 15 in https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf for this defintion.

My question is how to obtain the $a$-function for the symmetric group as a Weyl group via SAGE.

A more explicit formula in this case is given as follows: Let $x$ be a permutation corresponding to the pair of tableau $(P(x),Q(x))$ by the Robinson-Schendsted correspondence and $shape(Q(x)')=( \lambda_1,...,\lambda_k)$ where $Q(x)'$ is the transposed tableaux. Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$, see for example exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups". Groups".

Given a Weyl group $W$, Lusztig's $a$-function from W to the natural numbers is defined as a function constant on two-sided cells and $a(w)=l(w)-2 d(w)$ when $w$ is an involution, where $l(w)$ is the length function and $d(w)$ is the degree of the Kazhdan-Lusztig polynomial $P_{1,w}$. Since every two-sided cell contains an involution the function a is defined on all of W. See for example before conjecture 15 in https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf for this defintion.

My question is how to obtain the $a$-function for the symmetric group as a Weyl group via SAGE.

A more explicit formula in this case is given as follows: Let $x$ be a permutation corresponding to the pair of tableau $(P(x),Q(x))$ by the Robinson-Schendsted correspondence and $shape(Q(x)')=( \lambda_1,...,\lambda_k)$ where $Q(x)'$ is the transposed tableaux. Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$, see for example exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups".

Given a Weyl group $W$, Lusztig's $a$-function from W to the natural numbers is defined as a function constant on two-sided cells and $a(w)=l(w)-2 d(w)$ when $w$ is an involution, where $l(w)$ is the length function and $d(w)$ is the degree of the Kazhdan-Lusztig polynomial $P_{1,w}$. Since every two-sided cell contains an involution the function a is defined on all of W. See for example before conjecture 15 in https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf for this defintion.definition.

My question is how to obtain the $a$-function for the symmetric group as a Weyl group via SAGE. Sage.

A more explicit formula in this case is given as follows: follows: Let $x$ be a permutation corresponding to the pair of tableau $(P(x),Q(x))$ tableaux $(P(x),Q(x))$ by the Robinson-Schendsted Robinson-Schensted correspondence and $shape(Q(x)')=( and $\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$ where $Q(x)'$ is the transposed tableaux. tableau. Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$, see for example example exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups".

The values of the function can be seen for example for the symmetric group group on 3 symbols on top of page 323 of https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf .

Given a Weyl group $W$, Lusztig's $a$-function from W to the natural numbers is defined as a function constant on two-sided cells and $a(w)=l(w)-2 d(w)$ when $w$ is an involution, where $l(w)$ is the length function and $d(w)$ is the degree of the Kazhdan-Lusztig polynomial $P_{1,w}$. Since every two-sided cell contains an involution the function a is defined on all of W. See for example before conjecture 15 in https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf for this definition.

My question is how to obtain the $a$-function for the symmetric group as a Weyl group via Sage.

A more explicit formula in this case is given as follows: Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$ by the Robinson-Schensted correspondence and $\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$ where $Q(x)'$ is the transposed tableau. Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$, see for example exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups".

The values of the function can be seen for example for the symmetric group on 3 symbols on top of page 323 of https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf .

Given a Weyl group $W$, Lusztig's $a$-function from W to the natural numbers is defined as a function constant on two-sided cells and $a(w)=l(w)-2 d(w)$ when $w$ is an involution, where $l(w)$ is the length function and $d(w)$ is the degree of the Kazhdan-Lusztig polynomial $P_{1,w}$. Since every two-sided cell contains an involution the function a is defined on all of W. See for example before conjecture 15 in https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf for this definition.

My question is how to obtain the $a$-function for the symmetric group as a Weyl group via Sage.

A more explicit formula in this case is given as follows: Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$ by the Robinson-Schensted correspondence and $\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$ where $Q(x)'$ is the transposed tableau. Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$, see for example exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups".

The values of the function can be seen for example for the symmetric group on 3 symbols on top of page 323 of https://msp.org/pjm/2007/232-2/pjm-v232-n2-p06-s.pdf .