next | previous | forward | backward | up | index | toc

# BettiCharacters Example 4 -- a multigraded example

Consider the polynomial ring $\mathbb{Q} [x_1,x_2,y_1,y_2,y_3]$ with the variables $x_i$ of bidegree {1,0} and the variables $y_j$ of bidegree {0,1}. We consider the action of a product of two symmetric groups, the first permuting the $x_i$ variables and the second permuting the $y_j$ variables. We compute the Betti characters of this group on the resolution of the bigraded irrelevant ideal $\langle x_1,x_2\rangle \cap \langle y_1,y_2,y_3\rangle$. This is also the edge ideal of the complete bipartite graph $K_{2,3}$.

 i1 : R = QQ[x_1,x_2,y_1,y_2,y_3,Degrees=>{2:{1,0},3:{0,1}}] o1 = R o1 : PolynomialRing i2 : I = intersect(ideal(x_1,x_2),ideal(y_1,y_2,y_3)) o2 = ideal (x y , x y , x y , x y , x y , x y ) 2 3 1 3 2 2 1 2 2 1 1 1 o2 : Ideal of R i3 : RI = res I 1 6 9 5 1 o3 = R <-- R <-- R <-- R <-- R <-- 0 0 1 2 3 4 5 o3 : ChainComplex i4 : G = { matrix{{x_1,x_2,y_2,y_3,y_1}}, matrix{{x_1,x_2,y_2,y_1,y_3}}, matrix{{x_1,x_2,y_1,y_2,y_3}}, matrix{{x_2,x_1,y_2,y_3,y_1}}, matrix{{x_2,x_1,y_2,y_1,y_3}}, matrix{{x_2,x_1,y_1,y_2,y_3}} } o4 = {| x_1 x_2 y_2 y_3 y_1 |, | x_1 x_2 y_2 y_1 y_3 |, | x_1 x_2 y_1 y_2 y_3 ------------------------------------------------------------------------ |, | x_2 x_1 y_2 y_3 y_1 |, | x_2 x_1 y_2 y_1 y_3 |, | x_2 x_1 y_1 y_2 ------------------------------------------------------------------------ y_3 |} o4 : List i5 : A = action(RI,G) o5 = ChainComplex with 6 actors o5 : ActionOnComplex i6 : character A o6 = Character over R (0, {0, 0}) => | 1 1 1 1 1 1 | (1, {1, 1}) => | 0 2 6 0 0 0 | (2, {1, 2}) => | 0 -2 6 0 0 0 | (2, {2, 1}) => | 0 1 3 0 -1 -3 | (3, {1, 3}) => | 2 -2 2 0 0 0 | (3, {2, 2}) => | 0 -1 3 0 1 -3 | (4, {2, 3}) => | 1 -1 1 -1 1 -1 | o6 : Character

We can also compute the characters of some graded components of the quotient by this ideal.

 i7 : Q = R/I o7 = Q o7 : QuotientRing i8 : B = action(Q,G) o8 = QuotientRing with 6 actors o8 : ActionOnGradedModule i9 : character(B,{1,0}) o9 = Character over R (0, {1, 0}) => | 2 2 2 0 0 0 | o9 : Character i10 : character(B,{0,1}) o10 = Character over R (0, {0, 1}) => | 0 1 3 0 1 3 | o10 : Character i11 : character(B,{4,0}) o11 = Character over R (0, {4, 0}) => | 5 5 5 1 1 1 | o11 : Character i12 : character(B,{0,5}) o12 = Character over R (0, {0, 5}) => | 0 3 21 0 3 21 | o12 : Character

Note that all mixed degree components are zero.

 i13 : character(B,{1,1}) o13 = Character over R o13 : Character i14 : character(B,{2,1}) o14 = Character over R o14 : Character i15 : character(B,{1,2}) o15 = Character over R o15 : Character