## Thursday, October 6, 2016

### $F_4$ tensor products

The product of two irreducible representations of a simple Lie algebra can be decomposed into irreducible components. There are various techniques to calculate this decomposition, see for example chapter XIV in [1]. However, the decomposition can also be calculated by brute force.
From [1], page 117: We can calculate all the weights of the representations associated with highest weights $\Lambda_1$ and $\Lambda_2$, and using the Freudenthal recursion relations we can determine the multiplicity of each weight. Then the multiplicity of the weight $M$ in the product representation is found by writing $M = M_1 + M_2$, where $M_1$ and $M_2$ are weights of the two irreducible representations. The multiplicity of the weight $M$ in the product representation is $$\label{eq:20161006a} n_M = \sum_{M=M_1+M_2} n_{M_1} n_{M_2}$$ where $n_{M_1}$ and $n_{M_2}$ are the multiplicities of the weights $M_1$ and $M_2$. Now we know $\Lambda_1+\Lambda_2$ is the highest weight of one irreducible component so we can subtract its weights (with proper multiplicities) from the list. Now the highest remaining weight must be the highest weight of some irreducible component which again we find and eliminate from the list. Continuing this way, we exhaust the list of weights, and most likely, ourselves.
I have implemented this method in Mathematica. I test my calculations on tensor products in $F_4$. I have chosen the algebra $F_4$ because it is one of the exceptional Lie-algebras, so I know less about it than about the more familiar Lie-algebras. The rank is sufficiently high so that I do not want to try the brute force method by hand. On the other hand, it is sufficiently low so that a calculation with computer is still feasible. I checked my calculations against table 45 in Slansky [2]. Here is the table from Slansky.
 Source: Slansky, Group Theory for Unified Model Building, 1981

The results of my calculation are \begin{align*} 26 \times 26 =& 324 + 273 + 52 + 26 + 1 \\ 52 \times 26 =& 1053 + 273 + 26 \\ 52 \times 52 =& 1053^{\prime} + 1274 + 324 + 52 + 1 \\ 273 \times 26 =& 4096 + 1274 + 1053 + 324 + 273 + 52 + 26 \\ 273 \times 52 =& 8424 + 4096 + 1053 + 324 + 273 + 26 \\ 273 \times 273 =& 19448 + 19278 + 10829 + 8424 + 1053^{\prime} + 2652 + 2\times 4096 + 1274 + 2\times 1053 + \\ &2\times 324 + 2\times 273 + 52 + 26 + 1 \\ 324 \times 26 =& 2652 + 4096 + 1053 + 324 + 273 + 26 \\ 324 \times 52 =& 10829 + 4096 + 1274 + 324 + 273 + 52 \\ 324 \times 273 =& 34749 + 19278 + 10829 + 8424 + 2652 + 2\times 4096 + 1274 + 2\times 1053 + 324 + 2\times 273 + 52 + 26 \\ 324 \times 324 =& 16302 + 34749 + 19448 + 10829 + 8424 + 1053^{\prime} + 2652 + 2\times 4096 + 1274 + 1053 + 2\times 324 + 273 + 52 + 26 + 1 \\ 1053 \times 26 =& 10829 + 8424 + 1053^{\prime} + 4096 + 1274 + 1053 + 324 + 273 + 52 \\ 1053 \times 52 =& 17901 + 19278 + 8424 + 2652 + 4096 + 2\times 1053 + 273 + 26 \\ 1053 \times 273 =& 106496 + 29172 + 17901 + 34749 + 19448 + 19278 + 2\times 10829 + 2\times 8424 + 1053^{\prime} + 2652 + \\ &3\times 4096 + 2\times 1274 + 2\times 1053 + 2\times 324 + 2\times 273 + 52 + 26 \\ 1053 \times 324 =& 76076 + 106496 + 17901 + 34749 + 19448 + 2\times 19278 + 10829 + 2\times 8424 + 2652 + \\ &3\times 4096 + 1274 + 3\times 1053 + 324 + 2\times 273 + 26 \\ 1053 \times 1053 =& 160056 + 119119 + 12376 + 160056 + 107406 + 2\times 106496 + 2\times 29172 + 17901 + 16302 + \\ &2\times 34749 + 2\times 19448 + 2\times 19278 + 4\times 10829 + 3\times 8424 + 2\times 1053^{\prime} + 2652 + 4\times 4096 + 3\times 1274 + \\ &+2\times 1053 + 3\times 324 + 2\times 273 + 2\times 52 + 26 + 1 \\ 1053^{\prime} \times 26 =& 17901 + 8424 + 1053 \\ 1053^{\prime} \times 52 =& 12376 + 29172 + 10829 + 1053^{\prime} + 1274 + 52 \\ 1053^{\prime} \times 273 =& 119119 + 106496 + 17901 + 19278 + 10829 + 8424 + 4096 + 1053 + 273 \\ 1053^{\prime} \times 324 =& 160056 + 106496 + 29172 + 19448 + 10829 + 8424 + 1053^{\prime} + 4096 + 1274 + 324 \\ 1053^{\prime} \times 1053 =& 184756 + 379848 + 119119 + 107406 + 76076 + 106496 +\\ & 2\times 17901 + 34749 + 2\times 19278 + 2\times 8424 + 2652 + 4096 + 2\times 1053 + 273 + 26 \\ 1053^{\prime} \times 1053^{\prime} =& 100776 + 340119 + 226746 + 160056 + 12376 + 160056 + \\ &2\times 29172 + 16302 + 19448 + 10829 + 2\times 1053^{\prime} + 1274 + 324 + 52 + 1 \\ 1274 \times 26 =& 19278 + 8424 + 4096 + 1053 + 273 \\ 1274 \times 52 =& 29172 + 19448 + 10829 + 1053^{\prime} + 4096 + 1274 + 324 + 52 \\ 1274 \times 273 =& 107406 + 106496 + 17901 + 34749 + 19448 + 19278 + 10829 + 2\times 8424 + 2652 + \\ &2\times 4096 + 1274 + 2\times 1053 + 324 + 273 + 26 \\ 1274 \times 324 =& 160056 + 106496 + 29172 + 34749 + 19448 + 19278 + 2\times 10829 + 8424 + 1053^{\prime} + \\ &2\times 4096 + 2\times 1274 + 1053 + 324 + 273 + 52 \\ 1274 \times 1053 =& 379848 + 119119 + 205751 + 107406 + 76076 + 2\times 106496 + 2\times 17901 + \\ &2\times 34749 + 19448 + 3\times 19278 + 10829 + 3\times 8424 + 2\times 2652 + 3\times 4096 + 3\times 1053 + 324 + 2\times 273 + 26 \\ 1274 \times 1053^{\prime} =& 340119 + 420147 + 160056 + 12376 + 160056 + 106496 + 2\times 29172 + 34749 + 19448 +\\ & 2\times 10829 + 1053^{\prime} + 4096 + 2\times 1274 + 324 + 52 \\ 1274 \times 1274 =& 226746 + 420147 + 160056 + 12376 + 205751 + 160056 + 2\times 106496 + 2\times 29172 + 16302 + 34749 +\\ & 2\times 19448 + 19278 + 3\times 10829 + 8424 + 2\times 1053^{\prime} + 2652 + 2\times 4096 + 2\times 1274 + 2\times 324 + 273 + 52 + 1 \\ \end{align*} My results agree with Slansky's table. I have also extended Slansky's table with more products.
In Slansky's table there are also indices $s$ and $a$. I suppose they mean the symmetric product $v_i \otimes v_j+v_j \otimes v_i$ and the anti-symmetric product $v_i \otimes v_j-v_j \otimes v_i$ respectively. The brute force method described above can be adapted easily to cover this case. Instead of formula \eqref{eq:20161006a} I use for the symmetric product $$\label{eq:20161006b} n_M = \sum_{M=2 M_1} \frac{1}{2} n_{M_1} ( n_{M_1} + 1) + \frac{1}{2}\sum_{\substack{M=M_1+M_2\\ M_1 \neq M_2}} n_{M_1} n_{M_2}$$ and for the anti-symmetric product $$\label{eq:20161006c} n_M = \sum_{M=2 M_1} \frac{1}{2} n_{M_1} ( n_{M_1} - 1) + \frac{1}{2}\sum_{\substack{M=M_1+M_2\\ M_1 \neq M_2}} n_{M_1} n_{M_2}$$ When I use my Mathematica program with formula \eqref{eq:20161006b} and \eqref{eq:20161006c} I obtain
\begin{align*} 26 \times_s 26 =& 324 + 26 + 1 \\ 26 \times_a 26 =& 273 + 52 \\ 52 \times_s 52 =& 1053^{\prime} + 324 + 1 \\ 52 \times_a 52 =& 1274 + 52 \\ 273 \times_s 273 =& 19448 + 8424 + 1053^{\prime} + 2652 + 4096 + 1053 + 2\times324 + 26 + 1 \\ 273 \times_a 273 =& 19278 + 10829 + 4096 + 1274 + 1053 + 2\times273 + 52 \\ 324 \times_s 324 =& 16302 + 19448 + 8424 + 1053^{\prime} + 2652 + 4096 + 2\times324 + 26 + 1 \\ 324 \times_a 324 =& 34749 + 10829 + 4096 + 1274 + 1053 + 273 + 52 \\ 1053 \times_s 1053 =& 160056 + 107406 + 106496 + 29172 + 17901 + 16302 + 34749 + \\ &2\times19448 + 10829 + 2\times8424 + 2\times1053^{\prime} + 2652 + 2\times4096 + 1274 + 1053 + 3\times324 + 26 + 1 \\ 1053 \times_a 1053 =& 119119 + 12376 + 160056 + 106496 + 29172 + 34749 + 2\times19278 + \\ &3\times10829 + 8424 + 2\times4096 + 2\times1274 + 1053 + 2\times273 + 2\times52 \\ 1053^{\prime} \times_s 1053^{\prime} =& 100776 + 226746 + 160056 + 29172 + 16302 + 19448 + 2\times1053^{\prime} + 324 + 1 \\ 1053^{\prime} \times_a 1053^{\prime} =& 340119 + 12376 + 160056 + 29172 + 10829 + 1274 + 52 \\ 1274 \times_s 1274 =& 226746 + 160056 + 205751 + 106496 + 29172 + 16302 + 2\times19448 + 10829 + 8424 + \\ &2\times1053^{\prime} + 2652 + 4096 + 2\times324 + 1 \\ 1274 \times_a 1274 =& 420147 + 12376 + 160056 + 106496 + 29172 + 34749 + 19278 + 2\times10829 + 4096 + 2\times1274 + 273 + 52 \\ \end{align*} This again agrees with Slansky's table.
Remark
For high dimensional representations, the calculation with the brute force method takes a long time. For example, the calculation of the product $351\times 351$ in $E_6$ takes around 10 minutes. The reason is that the dimension of the product space is around 90000, and the calculation has loops with looking up values in this list etc. In Lie-ART [3], product decompositions are calculated with the more efficient Klimyk’s formula.
References
[1] Cahn, Semi-Simple Lie Algebras and Their Representations
[2] Slansky, Group Theory for Unified Model Building, 1981
[3] LieART – A Mathematica Application for Lie Algebras and Representation