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
o5 = {v v , 2v v v + v v , v v + 2v v v , 2v v v + v v , v 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
------------------------------------------------------------------------
2 2 2 2
v v v + v v v , 2v v v + v v , v v + 2v v v , v v + 2v v v , v v }
1 0 2 2 0 1 1 1 2 2 1 0 2 2 0 2 1 2 2 1 2 2 2
o5 : List
|
i6 : D#1
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 -
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 2 2
v v v , v v v - v v , v v - v v v , v v - v v v }
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
o11 = {x x , 3x x x + 2x x x , 3x x x + 6x x x x + x x , 3x x x +
0 0 0 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 2
-----------------------------------------------------------------------
3 2 3 2 2 2 2 2 2
2x x x , x x + 6x x x x + 3x x x , 3x x x x + 3x x x x + 3x x x x
0 2 0 0 1 0 1 0 1 1 0 1 0 0 1 2 0 1 0 2 0 2 0 1
-----------------------------------------------------------------------
3 3 2 2 2 2 2 2 2
+ x x x , 2x x x + 3x x x , x x x + 4x x x x x + x x x + 2x x x x
1 2 0 0 1 1 1 0 1 0 1 2 0 1 0 1 2 1 0 2 0 2 0 1
-----------------------------------------------------------------------
2 2 2 2 2 3 2 3 2 2
+ 2x x x x , 3x x x + 6x x x x + x x , x x , 3x x x x + 3x x x x +
1 2 0 1 0 0 2 0 2 0 2 2 0 1 1 0 1 1 2 1 0 1 2
-----------------------------------------------------------------------
3 2 2 2 2 2
x x x + 3x x x x , x x x + 2x x x x + 4x x x x x + 2x x x x +
0 2 1 1 2 0 1 0 1 2 0 1 0 2 0 2 0 1 2 1 2 0 2
-----------------------------------------------------------------------
2 2 2 2 3 2 2 2 2
x x x , 3x x x + 2x x x , 2x x x x + x x x + 2x x x x + 4x x x x x
2 0 1 1 1 2 1 2 1 0 1 1 2 1 0 2 0 2 1 2 1 2 0 1 2
-----------------------------------------------------------------------
2 2 2 3 2 2 2 2 2 2 2 3
+ x x x , x x + 6x x x x + 3x x x , 3x x x + 6x x x x + x x ,
2 0 1 0 2 0 2 0 2 2 0 2 1 1 2 1 2 1 2 2 1
-----------------------------------------------------------------------
3 2 2 2 2 3 2 2 2
x x x + 3x x x x + 3x x x x + 3x x x x , x x + 6x x x x + 3x x x ,
0 1 2 0 2 1 2 1 2 0 2 2 0 1 2 1 2 1 2 1 2 2 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 3
2x x x + 3x x x , 2x x x + 3x x x , x x }
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
o12 = {x x x - x x x , 2x x x - x x x x - 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
-----------------------------------------------------------------------
2 2 2 2 2 3 2 2
x x x x - 2x x x , 2x x x x - x x x x - x x x , x x x x - x x x x ,
0 1 0 1 1 0 1 0 0 1 2 0 2 0 1 1 2 0 0 1 0 2 0 2 0 1
-----------------------------------------------------------------------
3 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 , 2x x x x x +
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
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 3 2
x x x - 2x x x x - x x x x , 2x x x - x x x x - x x , x x x x -
1 0 2 0 2 0 1 1 2 0 1 0 0 2 0 2 0 2 2 0 0 1 1 2
-----------------------------------------------------------------------
2 2 3 2 2 2
x x x x , 2x x x x - x x x - x x x x , 2x x x + 2x x x x x -
1 2 0 1 1 0 1 2 0 2 1 1 2 0 1 0 1 2 0 2 0 1 2
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2
3x x x x - x x x , 2x x x x - 2x x x x x + x x x x - x x x , x x x
1 2 0 2 2 0 1 0 1 0 2 0 2 0 1 2 1 2 0 2 2 0 1 1 1 2
-----------------------------------------------------------------------
3 2 2 2 2 2 2
- x x x , 2x x x x + x x x x - 2x x x x x - x x x , 2x x x -
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
3x x x x + 2x x x x x - x x x , x x + x x x x - 2x x x , 2x 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
-----------------------------------------------------------------------
2 2 3 3 2 2 2 2
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 ,
1 2 1 2 2 1 0 1 2 1 2 0 2 2 0 1 2 0 2 1 2 1 2 0 2
-----------------------------------------------------------------------
2 3 2 2 2 3 2 2 3 2 2
x x + x x x x - 2x x x , x x x - x x x , x x x - x x x }
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
o13 = {x x x - 2x x x x + x x , x x - 2x x x x + x x x , x 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
-----------------------------------------------------------------------
2 2 3 2 2 2 2 2 2
x x x x - x x x x + x x x , x x x - x x x - 2x x x x + 2x x x x ,
0 1 0 2 0 2 0 1 1 2 0 0 1 2 1 0 2 0 2 0 1 1 2 0 1
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 3
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 ,
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 2 3 2 2 2 2 2
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 ,
0 1 1 2 1 0 1 2 0 2 1 1 2 0 1 0 1 2 0 2 0 1 2 2 0 1
-----------------------------------------------------------------------
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 -
0 1 0 2 0 2 0 1 2 1 2 0 2 2 0 1 0 1 1 2 0 2 1 2
-----------------------------------------------------------------------
2 2 2 2 2 2 2 3 2
x x x x x + x x x , x x x - 2x x x x x + x x x , x x - 2x x x x +
1 2 0 1 2 2 0 1 1 0 2 1 2 0 1 2 2 0 1 0 2 0 2 0 2
-----------------------------------------------------------------------
2 2 2 2 2 2 3 3 2 2
x x x , x x x - 2x x x x + x x , x x x - x x x x - x x x x +
2 0 2 1 1 2 1 2 1 2 2 1 0 1 2 0 2 1 2 1 2 0 2
-----------------------------------------------------------------------
2 2 3 2 2 2
x x x x , x x - 2x x x x + x x x }
2 0 1 2 1 2 1 2 1 2 2 1 2
o13 : List
|