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A finite set of the Euclidean space is called an $s$-distance set providedthe number of Euclidean distances in the set is $s$. Determining the largestpossible $s$-distance set for the Euclidean space of a given dimension ischallenging. This problem was solved only when dealing with small values of $s$and dimensions. Lison\v{e}k (1997) achieved the classification of the largest2-distance sets for dimensions up to $7$, using computer assistance and graphrepresentation theory. In this study, we consider a theory analogous to theseresults of Lison\v{e}k for the pseudo-Euclidean space $\mathbb{R}^{p,q}$. Weconsider an $s$-indefinite-distance set in a pseudo-Euclidean space that usesthe value \[ || x-y ||=(x_1-y_1)^2 +\cdots +(x_p-y_p)^2-(x_{p+1}-y_{p+1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \] instead of theEuclidean distance. We develop a representation theory for symmetric matricesin the context of $s$-indefinite-distance sets, which includes or improves theresults of Euclidean $s$-distance sets with large $s$ values. Moreover, weclassify the largest possible $2$-indefinite-distance sets for smalldimensions.

Embedding dimensions of matrices whose entries are indefinite distances in the pseudo-Euclidean space