Representing Finite Groups by Ambar N. Sengupta

Author:Ambar N. Sengupta
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(6.55)

Conclusion 1, stated in terms of the row and column partitions, says that Rows (sT) and Cols (T) are Young complements of each other if and only if s ∈ CTRT.

Proof. The condition that there does not exist two elements that are in one row of T′ = sT and also in one column of T means that

which, since T′ and T have the same shape, means that Rows(T′) is a Young complement of Cols(T). From Theorem 6.3, Rows(T′) is a Young complement for Cols(T) if and only if s1Rows(T′) = Rows(T) for some s1 ∈ FixCols(T). Since Rows(T′) = sRows(T), the condition is thus equivalent to the following:

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