This function computes the decomposition in irreducible $SL(n+1)$-representations of the tensor product $S^aV \otimes S^bV$, where $V = <v_0,\ldots,v_n>$ and $a \leq b$.
If $n = 1$, the decomposition is
$S^aV \otimes S^bV = S^{a+b}V \oplus S^{a+b-2}V \oplus S^{a+b-4}V \oplus \dots \oplus S^{b-a}V$,
while if $n > 1$, the decomposition is
$S^aV \otimes S^bV = S^{a+b}V \oplus V_{(a+b-2)\lambda_1 + \lambda_2} \oplus V_{(a+b-4)\lambda_1 + 2\lambda_2} \oplus \dots \oplus V_{(b-a)\lambda_1 + a\lambda_2}$,
where $\lambda_1$ and $\lambda_2$ are the two greatest fundamental weights of the Lie group $SL(n+1)$ and $V_{i\lambda_1+j\lambda_2}$ is the irreducible representation of highest weight $i\lambda_1+j\lambda_2$.
i1 : n = 2 o1 = 2 |
i2 : a = 1, b = 2 o2 = (1, 2) o2 : Sequence |
i3 : D = slIrreducibleRepresentationsTensorProduct(n,a,b); |
i4 : #D o4 = 2 |
i5 : D#0 2 2 2 2 2 2 2 2 o5 = {v v , 2v v v + v v , v v + 2v v v , 2v v v + v v , v v , v v v + v v v + v v v , 2v v v + v v , v v + 2v v v , v v 0 0 0 0 1 1 0 0 1 1 0 1 0 0 2 2 0 1 1 0 1 2 1 0 2 2 0 1 1 1 2 2 1 0 2 2 0 2 1 2 ---------------------------------------------------------------------------------------------------------------------------- 2 + 2v v v , v v } 2 1 2 2 2 o5 : List |
i6 : D#1 2 2 2 2 2 2 o6 = {v v v - v v , v v - v v v , v v v - v v , v v v - v v v , v v v - v v v , v v v - v v , v v - v v v , v v - v v v } 0 0 1 1 0 0 1 1 0 1 0 0 2 2 0 0 1 2 2 0 1 1 0 2 2 0 1 1 1 2 2 1 0 2 2 0 2 1 2 2 1 2 o6 : List |
If a polynomial ring R is given, then n = numgens R - 1 and $V = <R_0,\ldots,R_n>$.
i7 : R = QQ[x_0,x_1,x_2]; |
i8 : a = 2, b = 3 o8 = (2, 3) o8 : Sequence |
i9 : D = slIrreducibleRepresentationsTensorProduct(R,a,b); |
i10 : #D o10 = 3 |
i11 : D#0 2 3 2 2 3 2 2 2 2 3 2 2 3 2 3 2 2 2 2 2 o11 = {x x , 3x x x + 2x x x , 3x x x + 6x x x x + x x , 3x x x + 2x x x , x x + 6x x x x + 3x x x , 3x x x x + 3x x x x 0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 2 0 2 0 0 1 0 1 0 1 1 0 1 0 0 1 2 0 1 0 2 --------------------------------------------------------------------------------------------------------------------------- 2 3 3 2 2 2 2 2 2 2 2 2 2 2 2 3 + 3x x x x + x x x , 2x x x + 3x x x , x x x + 4x x x x x + x x x + 2x x x x + 2x x x x , 3x x x + 6x x x x + x x , 0 2 0 1 1 2 0 0 1 1 1 0 1 0 1 2 0 1 0 1 2 1 0 2 0 2 0 1 1 2 0 1 0 0 2 0 2 0 2 2 0 --------------------------------------------------------------------------------------------------------------------------- 2 3 2 2 3 2 2 2 2 2 2 2 2 2 3 x x , 3x x x x + 3x x x x + x x x + 3x x x x , x x x + 2x x x x + 4x x x x x + 2x x x x + x x x , 3x x x + 2x x x , 1 1 0 1 1 2 1 0 1 2 0 2 1 1 2 0 1 0 1 2 0 1 0 2 0 2 0 1 2 1 2 0 2 2 0 1 1 1 2 1 2 1 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 3 2x x x x + x x x + 2x x x x + 4x x x x x + x x x , x x + 6x x x x + 3x x x , 3x x x + 6x x x x + x x , x x x + 0 1 1 2 1 0 2 0 2 1 2 1 2 0 1 2 2 0 1 0 2 0 2 0 2 2 0 2 1 1 2 1 2 1 2 2 1 0 1 2 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 3 2 2 2 3 2 2 3 2 2 2 3 3x x x x + 3x x x x + 3x x x x , x x + 6x x x x + 3x x x , 2x x x + 3x x x , 2x x x + 3x x x , x x } 0 2 1 2 1 2 0 2 2 0 1 2 1 2 1 2 1 2 2 1 2 0 2 2 2 0 2 1 2 2 2 1 2 2 2 o11 : List |
i12 : D#1 2 2 3 2 2 2 2 3 2 2 3 2 3 2 2 2 2 2 3 o12 = {x x x - x x x , 2x x x - x x x x - x x , x x x - x x x , x x + x x x x - 2x x x , 2x x x x - x x x x - x x x , 0 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 2 0 2 0 0 1 0 1 0 1 1 0 1 0 0 1 2 0 2 0 1 1 2 0 --------------------------------------------------------------------------------------------------------------------------- 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 x x x x - x x x x , x x x - x x x , x x x - x x x + x x x x - x x x x , 2x x x x x + x x x - 2x x x x - x x x x , 0 1 0 2 0 2 0 1 0 1 1 1 0 1 0 1 2 1 0 2 0 2 0 1 1 2 0 1 0 1 0 1 2 1 0 2 0 2 0 1 1 2 0 1 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 3 2 2 2 3 2 2 2 2 2 2 2x x x - x x x x - x x , x x x x - x x x x , 2x x x x - x x x - x x x x , 2x x x + 2x x x x x - 3x x x x - x x x , 0 0 2 0 2 0 2 2 0 0 1 1 2 1 2 0 1 1 0 1 2 0 2 1 1 2 0 1 0 1 2 0 2 0 1 2 1 2 0 2 2 0 1 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 3 2 2 2 2 2 2 2x x x x - 2x x x x x + x x x x - x x x , x x x - x x x , 2x x x x + x x x x - 2x x x x x - x x x , 2x x x - 0 1 0 2 0 2 0 1 2 1 2 0 2 2 0 1 1 1 2 1 2 1 0 1 1 2 0 2 1 2 1 2 0 1 2 2 0 1 1 0 2 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 3 2 2 2 2 2 2 2 3 3 2 2 3x x x x + 2x x x x x - x x x , x x + x x x x - 2x x x , 2x x x - x x x x - x x , x x x + x x x x - 2x x x x , 0 2 1 2 1 2 0 1 2 2 0 1 0 2 0 2 0 2 2 0 2 1 1 2 1 2 1 2 2 1 0 1 2 1 2 0 2 2 0 1 2 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 3 2 2 2 3 2 2 3 2 2 x x x x - x x x x , x x + x x x x - 2x x x , x x x - x x x , x x x - x x x } 0 2 1 2 1 2 0 2 1 2 1 2 1 2 2 1 2 0 2 2 2 0 2 1 2 2 2 1 2 o12 : List |
i13 : D#2 2 2 2 2 3 2 3 2 2 2 2 2 2 3 2 2 2 2 2 o13 = {x x x - 2x x x x + x x , x x - 2x x x x + x x x , x x x x - x x x x - x x x x + x x x , x x x - x x x - 2x x x x 0 0 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 2 0 1 0 2 0 2 0 1 1 2 0 0 1 2 1 0 2 0 2 0 1 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 3 2 2 3 2 + 2x x x x , x x x x x - x x x - x x x x + x x x x , x x x - 2x x x x + x x , x x x x - x x x x - x x x + x x x x , 1 2 0 1 0 1 0 1 2 1 0 2 0 2 0 1 1 2 0 1 0 0 2 0 2 0 2 2 0 0 1 1 2 1 0 1 2 0 2 1 1 2 0 1 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 2 2 2 x x x - 2x x x x x + x x x , x x x x - x x x x x - x x x x + x x x , x x x x - x x x x - x x x x x + x x x , x x x 0 1 2 0 2 0 1 2 2 0 1 0 1 0 2 0 2 0 1 2 1 2 0 2 2 0 1 0 1 1 2 0 2 1 2 1 2 0 1 2 2 0 1 1 0 2 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 3 2 2 2 2 2 2 2 3 3 2 2 2 2 3 - 2x x x x x + x x x , x x - 2x x x x + x x x , x x x - 2x x x x + x x , x x x - x x x x - x x x x + x x x x , x x 1 2 0 1 2 2 0 1 0 2 0 2 0 2 2 0 2 1 1 2 1 2 1 2 2 1 0 1 2 0 2 1 2 1 2 0 2 2 0 1 2 1 2 --------------------------------------------------------------------------------------------------------------------------- 2 2 2 - 2x x x x + x x x } 1 2 1 2 2 1 2 o13 : List |
The object slIrreducibleRepresentationsTensorProduct is a method function.