By Tammo tom Dieck

This e-book is a jewel– it explains very important, important and deep subject matters in Algebraic Topology that you just won`t locate somewhere else, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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The same happen to the A4 triplet which splits into two triplets of S4 . On the other hand, the complex A4 one-dimensional irrep and its conjugate assemble into one real S4 doublet. 52 Finite groups: representations This is a general pattern: some irreps duplicate, and some assemble in going from the subgroup to the group. There is a systematic way to understand these patterns in terms of the automorphism group of A4 . Some mappings of the group into itself, like conjugation, live inside the group; they are the inner automorphisms.

76) 44 Finite groups: representations leads to two solutions, t = −1 and t = 3. The solution t = −1 yields by orthogonality w + w = 0. The second solution yields w + w = wz + wz = wz + wz = −12, which is not consistent unless w + w vanishes. Hence the one solution t = −1, w = 0 completes the character table. The characters of 11 and 12 are complex conjugates of one another: these two irreps are said to be conjugates as well, 12 = 11 . The three non-trivial classes are C2[3] (b, ab, aba, ba), C3[3] (b2 , ab2 , b2 a, ab2 a), and C4[2] (a, b2 ab, bab2 ), with the order of their elements in a square-bracketed superscript.

Setting Rβ = Rα , we find 1 n [α] −1 Mi[α] j (g)M pq (g ) = g 1 δ δ . 22) The (properly normalized) matrix elements of an irrep form an orthonormal set. This equation can be verified by multiplying both sides by an arbitrary matrix t j p , and summing over j and p, reproducing the result of Schur’s second lemma. The proper normalization is determined by summing over i and q. 23) g where the sum is over all elements of G. It is group-invariant by construction, in the sense that for any element ga , ( i(ga ), j (ga ) ) = = 1 n 1 n i(ga g) | j (ga g) g i(g ) | j (g ) = ( i, j ).