This classic monograph on representation theory and the symmetric group is suitable for advanced undergraduates and graduate students of mathematics. The Bulletin of the American Mathematical Society hailed Daniel Edwin Rutherford's treatment as `a long overdue account of Young's representation theory of the symmetric group,` noting that `many of Young's complicated proofs have been simplified through a free use of mathematical induction.` Starting with an elementary overview of the theory of the symmetric group, the book proceeds to an examination of Young's formula, followed by an introduction to the fundamental notion of a standard diagram. Subsequent chapters present the passage to orthogonal and natural representations. No prior knowledge of the general theory of group characters is assumed, and the author deduces the necessary theory for the symmetric group. The final chapter illustrates the substitutional aspect of the theory, and a helpful appendix and bibliography conclude the text.
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Argument. The crucial result of this chapter is theorem 12, which states that n! linearly independent substitutional expressions
can be constructed such that any substitutional expression whatever can be expressed as a linear combination of the units
and such that the relations
which employ the Kronecker delta, are satisfied. As is indicated by the notation, each
is associated with a certain shape Ī± and with two tableaux
and
of this shape. Now in Chapter II we defined n! different tableaux of each shape, so that without further qualification we should have far more tableaux than are required. The concept of a standard tableau is accordingly introduced. Of these there are a certain number fĪ± associated with each shape Ī±; and it is shown that
When our attention is restricted to standard tableaux only, it is found that we have exactly n! expressions
as is required by theorem 12.
If we can associate with the permutations Ļi, Ī±j ā¦ of the symmetric group a set of matrices UĻj, UĻj, ā¦ such that
UĻjUĻj = UĻjĻj,
the set of matrices is said to form a representation of the group. It is shown that the relations
mentioned above lead to one such representation for every partition Ī± and that, furthermore, each of these representations is irreducible. Such matrices, besides being of fundamental theoretical importance, have practical applications in the analysis of symmetry and in the solution of substitutional equations.
The irreducible representation associated with a par...
