On the Smallest Degrees of Projective Representations of the Groups PSL(n, q)

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Abstract
In this paper, we obtain information about the minimal degree δ of any non-trivial projective representation of the group PSL(n, q) with n ≧ 2 over an arbitrary given field K. Our main results for the groups PSL(n, q) (Theorems 4.2, 4.3, and 4.4) state that, apart from certain exceptional cases with small n, we have the following rather surprising situation: if q = pf (where p is a prime integer) and char K = p, then δ = n, but if q = pf and char K ≠ p, then δ is of a considerably higher order of magnitude, namely, δ is at least qn–l – 1 or if n = 2 and q is odd. Note that for n = 2, this lower bound for δ is the best possible. However, for n ≧ 3, this lower bound can conceivably be improved.

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