From 8fb7a195356ea7ee6dc828ee8fbc941d021b0b53 Mon Sep 17 00:00:00 2001 From: Federico Galetto Date: Sat, 15 Jun 2024 14:13:09 -0400 Subject: [PATCH] simplified example for Semidirect option --- BettiCharacters.m2 | 21 ++++++++++++++------- 1 file changed, 14 insertions(+), 7 deletions(-) diff --git a/BettiCharacters.m2 b/BettiCharacters.m2 index 510c67d..b77e24c 100644 --- a/BettiCharacters.m2 +++ b/BettiCharacters.m2 @@ -3254,19 +3254,26 @@ Node As a representative of this orbit we choose the unique degree $d$ whose entries are sorted in nonincreasing order from left to right; this can be obtained with the function @TO "rsort"@. + + We illustrate this use case. First, consider the action + on the polynomial ring. Example R = QQ[x_1..x_4,Degrees=>{{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}}] S4 = symmetricGroupActors R A = action(R,S4,Semidirect=>{uniquePermutations,rsort}) - actors(A,{2,0,0,0}) - character(A,{2,0,0,0}) - actors(A,{1,1,0,0}) - character(A,{1,1,0,0}) - oo == character(A,{1,0,1,0}) - I = ideal apply(subsets(gens R,3),product) + ideal apply(gens R, y -> y^5) + actors(A,{1,1,1,0}) + character(A,{1,1,1,0}) + Text + As expected, the character is the same if we compute it + for a different degree in the same orbit. + Example + oo == character(A,{1,0,1,1}) + Text + Next, consider the quotient by an ideal stable under the group action. + Example + I = ideal apply(subsets(gens R,3),product) M = R/I B = action(M,S4,Semidirect=>{uniquePermutations,rsort}) - actors(B,{2,1,0,0}) character(B,{2,1,0,0}) Text Similarly, the @TO "Semidirect"@ option can be used