# Sum of series involving projectors on eigenspaces of Hamming association scheme

I am trying to understand this paper but I am stuck at this step. Can someone please help?

Let $J_q$ be the $q\times q$ all ones matrix. Consider an orthonormal eigenbasis $\{e_0,e_1,\ldots,e_{n-1}\}$ of $J_q$ where $e_0 = \frac{1}{\sqrt q} (1,\ldots,1)$.

Consider tensor products of the form $v=e_{v_1}\otimes \dots \otimes e_{v_n}$. The weight of $v$ is the number of $v_i$'s that are not $0$. Let $E^{(n)}_k$ be the projector on the space spanned by vectors of the above form having weight $k$. Also, let $F_0=e_0 (e_0 \otimes e_0)^*$.

Let $G=F_0\otimes E^{(n-2)}_k$. For ${a,b}\subset[n]^2$, let $G_{a,b}$ be the matrix obtained from $G$ by permuting the first and second component of the tensor product with the $a^{th}$ and $b^{th}$ components respectively.

I want to show that $$\sum_{(a,b),(c,d)}G_{a,b}^* G_{c,d} = \frac{(n-k)(n-k+1)}{2}E^{(n)}_k.$$ Here the sum varies over all $a<b$ and $c<d$.

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Did you really mean orthonormal basis of $J_q$? All rows of $J_q$ are multiples of $e_0$. –  Will Orrick Apr 29 '12 at 13:45
Thanks. I meant the orthonormal eigenbasis of $J_q$. –  Shitikanth Apr 29 '12 at 13:54