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The q-analog of the Kramer-Mesner matrix


The next step in the construction is the evaluation of the entries of the Kramer-Mesner matrices. We achieve this goal by using the information obtained during the ladder game.

We can restrict attention to Kramer-Mesner matrices $M$, because of

\begin{displaymath}M_{t}\cdot M_{k} =
\left\vert\left[ k-t\atop k-t'\right]_q\right\vert\cdot M_{k}.\end{displaymath}


(The analogous equation -- with binomial coefficients instead of the Gaussian numbers -- holds for designs on sets, the proof of it is easily generalized to the $q$-analog.

The matrix $M$can be obtained as follows: We assume the representatives of the double cosets of $A$and $GL_v(q)_{S_{t}}$, representatives of the double cosets of $A$and $GL_v(q)_{S_t}$and the corresponding orbit graph. Now the entry of $M_{$indexed by the orbits

\begin{displaymath}\varphi^{-1}_t(A\Phi GL_v(q)_{S_t})\in A{\backslash\!\!\backs...
...{t}})\in A{\backslash\!\!\backslash}\left[v\atop t\right]_q\end{displaymath}


is

\begin{displaymath}\sum\frac {\vert A_{\Phi GL_v(q)_{S_t}}\vert}{\vert A_{\Gamma (GL_v(q)_{S_{t}}\cap
GL_v(q)_{S_t})}\vert},\end{displaymath}


where the sum is over all double cosets $A\Gamma(GL_v(q)_{S_{t}}\cap
GL_v(q)_{S_t})$which are connected with the double coset $A\Psi
GL_v(q)_{S_{t}}$and which are connected with the double coset $A\Phi GL_v(q)_{S_t}$at the same time.


If we take our last example then we get the Kramer-Mesner matrix:

$M_{1,2}^{M_4(2)}$

(3,0,4)

(3,1,2,)

(3,2,6)

(3,3,6)

(3,4,8)

(3,5,4)

(5,0,6)

6/2

6/2

0

6/6

0

0

(5,1,4)

4/4

4/2

4/2

0

4/4

4/4

(5,2,6)

0

6/2

0

6/6

0

6/2

(5,3,24)

0

0

0

24/6

24/8

0

,

and thus the $2$-analog of the Kramer-Mesner matrix turns out to be

\begin{displaymath}
M_{1,2}^{M_4(2)}=
\begin{array}{c}
3 3 0 1 0 0\\
1 2 2 0 1 1\\
0 3 0 1 0 3\\
0 0 0 4 3 0\\
\end{array}.
\end{displaymath}



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