Given the basis vectors of a 10-dimensional representation of $SO(10)$, how can I compute the basis vectors of the 54-dimensional representation?

Question

Because $10 \otimes 10 = 1_s \oplus 54_s \oplus45_a$ we can write each element of $54$ as a $10×10$ matrix.

The usual basis vectors of the 10-dim rep are $$ \begin{pmatrix}1 \\0 \\ \vdots \end{pmatrix} \quad \begin{pmatrix}0 \\1 \\ \vdots \end{pmatrix} \quad \ldots$$

How can I use this to compute the basis vectors of the $54$ dimensional representation, written as $10 \times 10 $ matrix?

dear all, maybe you know this https://math.stackexchange.com/q/4220583/955245 — Марина Marina S, Aug 10 '21 at 13:14

score 1 · Answer 1 · answered Aug 05 '15 at 13:32

1

Simply use the Kronecker product $e_i \otimes e_j$ to compute the basis vectors, which are then matrices for the product representation. Then symmetrize and antisymmetrize to get basis vectors for the corresponding irreducible representations.

answered Aug 05 '15 at 13:32

jak

1,596

Well if you can answer this explicitly https://math.stackexchange.com/q/4214397/141334 then I believe that what you said would work – annie marie cœur Aug 01 '21 at 18:39

Given the basis vectors of a 10-dimensional representation of $SO(10)$, how can I compute the basis vectors of the 54-dimensional representation?

1 Answers1