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REPRESENTATIONS of the SYMMETRIC GROUP and POLYNOMIAL IDENTITIES These Are the Lecture Notes from a Short Course Given by the Au REPRESENTATIONS OF THE SYMMETRIC GROUP AND POLYNOMIAL IDENTITIES LUIZ A. PERESI These are the lecture notes from a short course given by the author during the CIMPA Research School on Associative and Nonassociative Algebras and Dialge- bras: Theory and Algorithms - In Honour of Jean-Louis Loday (1946-2012), held at CIMAT, Guanajuato, Mexico, February 17 to March 2, 2013. Abstract. Let Sn denote the symmetric group on n symbols. When F has characteristic zero or greater than n, the group algebra FSn is a direct sum of p(n) matrix algebras over F, where p(n) is the number of partitions of n. We present an efficient method due to J. M. Clifton (1981) that calculates the matrix associated to each element of Sn, for each partition of n. In 1950, A. I. Malcev and W. Specht independently used the representation theory of the symmetric group to classify polynomial identities of algebraic structures. Starting in the 1970's the method was further developed by A. Regev (see the survey paper Regev [50]). The method was implemented as a computer algebra system by I. R. Hentzel in 1977. We present this computational method and give applications to research on identities of Cayley-Dickson algebras, nuclear elements of the free alternative algebra and special identities of Bol algebras. For some applications of the computational method to other algebraic structures see the papers [9]-[14] by M. R. Bremner and the author. 1. Representations of the symmetric group Let n be a positive integer, Sn the symmetric group on f1; : : : ; ng and FSn the group algebra over the field F. We remind that if G is a group then the group algebra FG is the vector space over F with basis G, and the multiplication in FG is given by extending the multiplication in G. The representation theory of FSn was determined in 1900 by Alfred Young (1873-1940). It has since been simplified and exposited by many authors. See for example Boerner [4] and James and Kerber [40]. We give an exposition of this theory based on Clifton [15]. In his review (MR0624907 (82j:20024)) of Clifton's paper, James wrote the following: \Most methods for working out the matrices for the natural representation are messy, but this paper gives an approach which is simple both to prove and to apply." 1.1. Frames and tableaus. A partition of n is an ordered sequence of positive integers λ = λ1 : : : λk satisfying λ1 ≥ λ2 ≥ · · · ≥ λk and λ1+···+λk = n. The frame corresponding to λ consists of n boxes in k left-justified rows of lengths λ1; : : : ; λk. For example, if n = 3 then there are 3 partitions 3; 21; 111 with corresponding frames 1 2 LUIZ A. PERESI A tableau corresponding to λ consists of a bijective assignment of the numbers 1; : : : ; n to the n boxes. A standard tableau is one in which the assigned numbers increase in each row from left to right and in each column from top to bottom. For example, if n = 3 then the standard tableaus are: 1 1 2 1 3 1 2 3 2 3 2 3 The number of standard tableaus corresponding to partition λ = λ1 : : : λk of n is given by Y n! (li − lj) i<j where li = λi + k − i (i = 1; : : : ; k): l1! l2! : : : lk! If T is a tableau we denote by T (i; j) the number in row i and column j. The lexicographical order on tableaus is defined by T < T 0 if and only if T (i; j) < T 0(i; j) where i is the least row index for which T (i; j) 6= T 0(i; j). For finite groups we have: Theorem 1.1. (H. Maschke 1899) For any finite group G and any field F, of characteristic 0 or characteristic p not dividing jGj, the group algebra FG is semisimple. It follows that FG is isomorphic to the direct sum of simple ideals, and each simple ideal is isomorphic to the endomorphism algebra of a vector space over a division ring over F. For G = Sn, these division rings are all equal to F and we have: Corollary 1.2. The Wedderburn decomposition of FSn is given by an iso- morphism M φ: F Sn ! Mdλ (F); λ where the summation is over all partitions λ of n, and dλ is the corresponding number of standard tableaus. A permutation π of f1; 2; : : : ; ng is denoted by the sequence π(1)π(2) : : : π(n). For n = 3 we have the isomorphism φ: FS3 ! F ⊕ M2(F) ⊕ F: given by 1 0 0 1 φ(123) = 1; ; 1 ; φ(132) = 1; ; −1 ; 0 1 1 0 1 −1 −1 1 φ(213) = 1; ; −1 ; φ(231) = 1; ; 1 ; 0 −1 −1 0 0 −1 −1 0 φ(312) = 1; ; 1 ; φ(321) = 1; ; −1 : 1 −1 −1 1 REPRESENTATIONS AND IDENTITIES 3 1.2. Computing the irreducible representations. Let λ be a partition of n. Let T1;:::; Tdλ be the standard tableaus for λ in lexicographical order. For π 2 Sn λ we denote by Rπ the dλ ×dλ matrix representing π, that is, the projection of π onto the summand Mdλ (F) in the Wedderburn decomposition given by Corollary 1.2.The λ λ λ map R : Sn ! Mdλ (F) defined by R (π) = Rπ is the irreducible representation of Sn corresponding to the partition λ. Let ι denote the identity permutation in Sn. The matrix Rλ(π) is given by λ λ −1 λ Rπ = (Aι ) Aπ; λ λ where Aι and Aπ are the Clifton matrices corresponding to ι and π. λ The Clifton matrix Aπ is the dλ × dλ matrix defined by the following algorithm λ for computing each entry (Aπ)ij: • Apply π to the standard tableau Tj obtaining the (possibly non-standard) tableau πTj. • If there exist two numbers that appear together both in a column of Ti and λ in a row of πTj, then (Aπ)ij = 0. • Otherwise, there exists a vertical permutation κ 2 Sn which { leaves the columns of Ti fixed as sets, { and takes the numbers of Ti into the rows they occupy in πTj. λ In this case, (Aπ)ij = (κ), the sign of κ. λ For n ≤ 4 and for all partitions λ, Aι is the identity matrix. For n ≥ 5 and for λ some partitions λ, the matrix Aι is not the identity matrix; however this matrix is always invertible. Example 1.3. Let n = 3 and λ = 21. We have that dλ = 2. For π = 213 we calculate Aπ. (i; j) Ti π(Tj) 1 2 2 1 (1; 1) T = πT = κ = ι (κ) = 1 1 3 1 3 1 2 2 3 (1; 2) T = πT = κ = 321 (κ) = −1 1 3 2 1 1 3 2 1 (2; 1) T = πT = κ does not exist 2 2 1 3 1 3 2 3 (2; 2) T = πT = κ = 213 (κ) = −1 2 2 2 1 1 −1 This gives the Clifton matrix Aλ = : π 0 −1 Example 1.4. Let n = 5 and λ = 32. In this case dλ = 5 and the standard tableaus are 1 2 3 1 2 4 1 2 5 1 3 4 1 3 5 4 5 3 5 3 4 2 5 2 4 4 LUIZ A. PERESI λ To calculate the (1; 5) entry of the Clifton matrix Aι , we consider T1 and ιT5 = T5. In this case the transposition κ = 15342 interchanging 2 and 5 is the required λ vertical permutation, so (Aι )15 = −1. In fact, we have 2 1 0 0 0 −1 3 2 1 0 0 0 1 3 6 0 1 0 0 0 7 6 0 1 0 0 0 7 λ 6 7 λ −1 6 7 A = 6 0 0 1 0 0 7 ; (A ) = 6 0 0 1 0 0 7 : ι 6 7 ι 6 7 4 0 0 0 1 0 5 4 0 0 0 1 0 5 0 0 0 0 1 0 0 0 0 1 λ This is the simplest case in which Aι is not the identity matrix. λ To illustrate the difference that can exist between the Clifton matrix Aπ and the λ λ −1 λ representation matrix Rπ = (Aι ) Aπ, consider the 5-cycle π = 23451; in this case we calculate λ λ −1 λ Rπ (Aι ) Aπ 2 −1 −1 1 1 0 3 2 1 0 0 0 1 3 2 −1 0 1 0 0 3 6 −1 0 0 0 1 7 6 0 1 0 0 0 7 6 −1 0 0 0 1 7 6 7 6 7 6 7 6 0 −1 0 0 0 7 = 6 0 0 1 0 0 7 6 0 −1 0 0 0 7 6 7 6 7 6 7 4 −1 0 0 1 0 5 4 0 0 0 1 0 5 4 −1 0 0 1 0 5 0 −1 0 1 0 0 0 0 0 1 0 −1 0 1 0 L The Wedderburn decomposition φ: F Sn ! λ Mdλ (F) has an inverse −1 M φ : Mdλ (F) ! F Sn λ −1 λ λ λ given by φ (Eij) = Uij, where λ is a partition of n and Eij is the dλ × dλ matrix λ unit. We describe how the elements Uij 2 FSn can be calculate. λ λ Let Ti be the i-th standard tableau in lexicographical order. We denote by Ri λ the subgroup of Sn which permutes the elements of Ti within each row but leaves λ the rows fixed as sets.
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