Identifier
Values
[+,+] => [1,2] => [2,1] => 0
[-,+] => [2,1] => [1,2] => 1
[+,-] => [1,2] => [2,1] => 0
[-,-] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 1
[+,+,+] => [1,2,3] => [3,2,1] => 0
[-,+,+] => [2,3,1] => [1,3,2] => 1
[+,-,+] => [1,3,2] => [2,3,1] => 1
[+,+,-] => [1,2,3] => [3,2,1] => 0
[-,-,+] => [3,1,2] => [2,1,3] => 1
[-,+,-] => [2,1,3] => [3,1,2] => 1
[+,-,-] => [1,2,3] => [3,2,1] => 0
[-,-,-] => [1,2,3] => [3,2,1] => 0
[+,3,2] => [1,3,2] => [2,3,1] => 1
[-,3,2] => [3,1,2] => [2,1,3] => 1
[2,1,+] => [2,3,1] => [1,3,2] => 1
[2,1,-] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [2,1,3] => 1
[3,1,2] => [2,3,1] => [1,3,2] => 1
[3,+,1] => [2,3,1] => [1,3,2] => 1
[3,-,1] => [3,1,2] => [2,1,3] => 1
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => 0
[-,+,+,+] => [2,3,4,1] => [1,4,3,2] => 1
[+,-,+,+] => [1,3,4,2] => [2,4,3,1] => 1
[+,+,-,+] => [1,2,4,3] => [3,4,2,1] => 1
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => 1
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => 2
[-,+,+,-] => [2,3,1,4] => [4,1,3,2] => 1
[+,-,-,+] => [1,4,2,3] => [3,2,4,1] => 1
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => 1
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,-,+] => [4,1,2,3] => [3,2,1,4] => 1
[-,-,+,-] => [3,1,2,4] => [4,2,1,3] => 1
[-,+,-,-] => [2,1,3,4] => [4,3,1,2] => 1
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => 0
[+,+,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[-,+,4,3] => [2,4,1,3] => [3,1,4,2] => 2
[+,-,4,3] => [1,4,2,3] => [3,2,4,1] => 1
[-,-,4,3] => [4,1,2,3] => [3,2,1,4] => 1
[+,3,2,+] => [1,3,4,2] => [2,4,3,1] => 1
[-,3,2,+] => [3,4,1,2] => [2,1,4,3] => 1
[+,3,2,-] => [1,3,2,4] => [4,2,3,1] => 1
[-,3,2,-] => [3,1,2,4] => [4,2,1,3] => 1
[+,3,4,2] => [1,4,2,3] => [3,2,4,1] => 1
[-,3,4,2] => [4,1,2,3] => [3,2,1,4] => 1
[+,4,2,3] => [1,3,4,2] => [2,4,3,1] => 1
[-,4,2,3] => [3,4,1,2] => [2,1,4,3] => 1
[+,4,+,2] => [1,3,4,2] => [2,4,3,1] => 1
[-,4,+,2] => [3,4,1,2] => [2,1,4,3] => 1
[+,4,-,2] => [1,4,2,3] => [3,2,4,1] => 1
[-,4,-,2] => [4,1,2,3] => [3,2,1,4] => 1
[2,1,+,+] => [2,3,4,1] => [1,4,3,2] => 1
[2,1,-,+] => [2,4,1,3] => [3,1,4,2] => 2
[2,1,+,-] => [2,3,1,4] => [4,1,3,2] => 1
[2,1,-,-] => [2,1,3,4] => [4,3,1,2] => 1
[2,1,4,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,1,+] => [3,4,1,2] => [2,1,4,3] => 1
[2,3,1,-] => [3,1,2,4] => [4,2,1,3] => 1
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 1
[2,4,1,3] => [3,4,1,2] => [2,1,4,3] => 1
[2,4,+,1] => [3,4,1,2] => [2,1,4,3] => 1
[2,4,-,1] => [4,1,2,3] => [3,2,1,4] => 1
[3,1,2,+] => [2,3,4,1] => [1,4,3,2] => 1
[3,1,2,-] => [2,3,1,4] => [4,1,3,2] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,+,1,+] => [2,3,4,1] => [1,4,3,2] => 1
[3,-,1,+] => [3,4,1,2] => [2,1,4,3] => 1
[3,+,1,-] => [2,3,1,4] => [4,1,3,2] => 1
[3,-,1,-] => [3,1,2,4] => [4,2,1,3] => 1
[3,+,4,1] => [2,4,1,3] => [3,1,4,2] => 2
[3,-,4,1] => [4,1,2,3] => [3,2,1,4] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[3,4,2,1] => [3,4,1,2] => [2,1,4,3] => 1
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => 1
[4,1,+,2] => [2,3,4,1] => [1,4,3,2] => 1
[4,1,-,2] => [2,4,1,3] => [3,1,4,2] => 2
[4,+,1,3] => [2,3,4,1] => [1,4,3,2] => 1
[4,-,1,3] => [3,4,1,2] => [2,1,4,3] => 1
[4,+,+,1] => [2,3,4,1] => [1,4,3,2] => 1
[4,-,+,1] => [3,4,1,2] => [2,1,4,3] => 1
[4,+,-,1] => [2,4,1,3] => [3,1,4,2] => 2
[4,-,-,1] => [4,1,2,3] => [3,2,1,4] => 1
[4,3,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[4,3,2,1] => [3,4,1,2] => [2,1,4,3] => 1
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Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
upper permutation
Description
The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $v$.