# To find $\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$

Let $$S_6$$ be the symmetric group of degree $$6$$ and $$F$$ be any finite field of characteristic $$2.$$ Then $$2$$-regular partition of $$6$$ are $$(5,1)$$, $$(4,2)$$ and $$(3,2,1)$$ . I have to find $$\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$$where $$S^{\mu}$$ is the Specht module corresponding to partition $$\mu.$$ I know $$\dim(D^{(5,1)})=\dim\frac{S^{(5,1)}}{S^{(5,1)}\cap S^{(5,1)\perp}}=4$$ as $$\dim S^{(n-1, 1)}=n-1$$ and $$S^{(n-1, 1)\perp}\subseteq S^{(n-1, 1)}$$ as $$n=6.$$ For more details see at page number $$18$$ of representation theory of symmetric groups by G.D.James book. But I am unable to find dimensions of $$D^{(4,2)}$$and $$D^{(3,2,1)}.$$ I think to find rank of Gram matrix is very lengthy process. Please help me to obtain these dimensions. Thanks.

• @Martin Sleziak Thanks – neelkanth Jan 9 '19 at 15:33