Representation theory of the symmetric group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. The symmetric group S has order "n"!. Its conjugacy classes are labeled by partitions of "n". Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions  of "n". Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of "n" or equivalently Young diagrams of size "n". Each such irreducible representation can in fact be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram. The dimension formula_1 of the representation that corresponds to the Young diagram formula_2  is given by the hook length formula. To each irreducible representation ρ we can associate an irreducible character, χ. To compute χ(π) where π is a permutation, one can use the combinatorial Murnaghan–Nakayama rule . Note that χ is constant on conjugacy classes, that is, χ(π) = χ(σπσ) for all permutations σ. Over other fields the situation can become much more complicated. If the field "K" has characteristic equal to zero or greater than "n" then by Maschke's theorem the group algebra "K"S is semisimple. In these cases the irreducible representations defined over the integers give the complete set  of irreducible representations (after reduction modulo the characteristic if necessary). However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of modules rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called "Specht modules", and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimensions  are not known in general. The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory. Low-dimensional representations. Symmetric groups. The lowest-dimensional representations of the symmetric groups can be described explicitly, and over arbitrary fields. The smallest two degrees in characteristic zero are described here: Every symmetric group has a one-dimensional representation called the trivial representation, where every element acts as the one by one identity matrix. For , there is another irreducible representation of degree 1, called the sign representation  or alternating character, which takes a permutation to the one by one matrix with entry ±1 based on the sign of the permutation. These are the only one-dimensional representations of the symmetric groups, as one-dimensional representations are abelian, and the abelianization of the symmetric group is C, the cyclic group of order 2. For all "n", there is an "n"-dimensional representation of the symmetric group of order "n!", called the , which consists of permuting "n" coordinates. This has the trivial subrepresentation consisting of vectors whose coordinates are all equal. The orthogonal complement consists of those vectors whose coordinates  sum to zero, and when , the representation on this subspace is an -dimensional irreducible representation, called the standard representation. Another -dimensional irreducible representation is found by tensoring with the sign representation. An exterior power formula_3 of the standard representation formula_4 is irreducible provided formula_5 . For , these are the lowest-dimensional irreducible representations of S – all other irreducible representations have dimension at least "n". However for , the surjection from S to S allows S to inherit a two-dimensional irreducible representation. For , the exceptional transitive embedding of S into S produces another pair of five-dimensional irreducible representations. Alternating groups. The  representation theory of the alternating groups is similar, though the sign representation disappears. For , the lowest-dimensional irreducible representations are the trivial representation in dimension one, and the -dimensional representation from the other summand of the permutation representation, with all other irreducible representations having higher dimension, but there are exceptions for smaller "n". The alternating groups for have only one one-dimensional irreducible representation, the trivial representation. For there are two additional one-dimensional irreducible representations, corresponding to maps to the cyclic group of order 3: and . Tensor products of representations. Kronecker coefficients. The tensor product of two representations of  formula_6 corresponding to the Young diagrams formula_7 is a combination of irreducible representations of formula_6, The coefficients formula_10 are called the Kronecker coefficients of the symmetric group. They can be computed from the characters of the representations : The sum is over partitions formula_12 of formula_13, with formula_14 the corresponding conjugacy classes. The values of the characters formula_15 can be computed using the Frobenius formula. The coefficients formula_16 are where formula_18 is the number of times formula_19 appears in formula_12, so that formula_21. A few examples, written in terms of Young diagrams : There is a simple rule for computing formula_26 for  any Young diagram formula_2 : the result is the sum of all Young diagrams that are obtained from formula_2 by removing one box and then adding one box, where the coefficients are one except for formula_2 itself, whose coefficient is formula_30, i.e., the number of different row lengths minus one. A constraint on the irreducible constituents of formula_31 is where the depth formula_33 of a Young diagram is the number of boxes that do not belong to the first row. Reduced Kronecker coefficients. For formula_2 a Young diagram and formula_35, formula_36 is a Young diagram of size formula_13. Then formula_38 is  a bounded, non-decreasing function of formula_13, and is called a reduced Kronecker coefficient or stable Kronecker coefficient. There are known bounds on the value of formula_13 where formula_38 reaches its limit. The reduced Kronecker coefficients are structure constants of Deligne categories of representations of formula_6 with formula_44. In contrast to Kronecker coefficients, reduced Kronecker coefficients are defined for any triple of Young diagrams, not necessarily of the same size. If formula_45, then formula_46 coincides with the Littlewood-Richardson coefficient formula_47. Reduced Kronecker coefficients can be written as linear combinations of Littlewood-Richardson coefficients via a change of bases in  the space of symmetric functions, giving rise to expressions that are manifestly integral although not manifestly positive. Reduced Kronecker coefficients can also be written in terms of Kronecker and Littlewood-Richardson coefficients formula_48 via Littlewood's formula Conversely, it is possible to recover the Kronecker coefficients as linear combinations of reduced Kronecker coefficients. Reduced Kronecker coefficients are implemented in the computer algebra system SageMath. Eigenvalues of complex representations. Given an element formula_50 of cycle-type formula_51 and order formula_52, the eigenvalues of formula_53 in a complex representation of formula_6 are of the type formula_55 with formula_56, where the integers formula_57  are called the cyclic exponents of formula_53 with respect to the representation. There is a combinatorial description of the cyclic exponents of the symmetric group (and wreath products thereof). Defining formula_59, let the formula_60-index of a standard Young tableau be the sum of the values of formula_61 over the tableau's descents, formula_62. Then the cyclic exponents of the representation of formula_6 described by the Young diagram formula_2 are the formula_60-indices of the corresponding Young tableaux. In particular, if formula_66 is of order formula_13, then formula_68, and formula_69 coincides with the major index of formula_70 (the sum of  the descents). The cyclic exponents of an irreducible representation of formula_6 then describe how it decomposes into representations of the cyclic group formula_72, with formula_55 being interpreted as the image of formula_53 in the (one-dimensional) representation characterized by formula_75.