Symmetric group

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Not to be confused with Symmetry group. Symmetric group_sentence_0

Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. Symmetric group_sentence_1

For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. Symmetric group_sentence_2

Definition and first properties Symmetric group_section_0

Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in (, Ch. Symmetric group_sentence_3

11), (, Ch. Symmetric group_sentence_4

8), and (). Symmetric group_sentence_5

Applications Symmetric group_section_1

The symmetric group on a set of size n is the Galois group of the general polynomial of degree n and plays an important role in Galois theory. Symmetric group_sentence_6

In invariant theory, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric functions. Symmetric group_sentence_7

In the representation theory of Lie groups, the representation theory of the symmetric group plays a fundamental role through the ideas of Schur functors. Symmetric group_sentence_8

In the theory of Coxeter groups, the symmetric group is the Coxeter group of type An and occurs as the Weyl group of the general linear group. Symmetric group_sentence_9

In combinatorics, the symmetric groups, their elements (permutations), and their representations provide a rich source of problems involving Young tableaux, plactic monoids, and the Bruhat order. Symmetric group_sentence_10

Subgroups of symmetric groups are called permutation groups and are widely studied because of their importance in understanding group actions, homogeneous spaces, and automorphism groups of graphs, such as the Higman–Sims group and the Higman–Sims graph. Symmetric group_sentence_11

Elements Symmetric group_section_2

The elements of the symmetric group on a set X are the permutations of X. Symmetric group_sentence_12

Multiplication Symmetric group_section_3

The group operation in a symmetric group is function composition, denoted by the symbol ∘ or simply by juxtaposition of the permutations. Symmetric group_sentence_13

The composition f ∘ g of permutations f and g, pronounced "f of g", maps any element x of X to f(g(x)). Symmetric group_sentence_14

Concretely, let (see permutation for an explanation of notation): Symmetric group_sentence_15

Applying f after g maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. Symmetric group_sentence_16

So composing f and g gives Symmetric group_sentence_17

A cycle of length L = k · m, taken to the k-th power, will decompose into k cycles of length m: For example, (k = 2, m = 3), Symmetric group_sentence_18

Verification of group axioms Symmetric group_section_4

To check that the symmetric group on a set X is indeed a group, it is necessary to verify the group axioms of closure, associativity, identity, and inverses. Symmetric group_sentence_19

Symmetric group_ordered_list_0

  1. The operation of function composition is closed in the set of permutations of the given set X.Symmetric group_item_0_0
  2. Function composition is always associative.Symmetric group_item_0_1
  3. The trivial bijection that assigns each element of X to itself serves as an identity for the group.Symmetric group_item_0_2
  4. Every bijection has an inverse function that undoes its action, and thus each element of a symmetric group does have an inverse which is a permutation too.Symmetric group_item_0_3

Transpositions Symmetric group_section_5

Main article: Transposition (mathematics) Symmetric group_sentence_20

A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Symmetric group_sentence_21

Every permutation can be written as a product of transpositions; for instance, the permutation g from above can be written as g = (1 2)(2 5)(3 4). Symmetric group_sentence_22

Since g can be written as a product of an odd number of transpositions, it is then called an odd permutation, whereas f is an even permutation. Symmetric group_sentence_23

The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. Symmetric group_sentence_24

There are several short proofs of the invariance of this parity of a permutation. Symmetric group_sentence_25

The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd. Symmetric group_sentence_26

Thus we can define the sign of a permutation: Symmetric group_sentence_27

With this definition, Symmetric group_sentence_28

is a group homomorphism ({+1, –1} is a group under multiplication, where +1 is e, the neutral element). Symmetric group_sentence_29

The kernel of this homomorphism, that is, the set of all even permutations, is called the alternating group An. Symmetric group_sentence_30

It is a normal subgroup of Sn, and for n ≥ 2 it has n!/2 elements. Symmetric group_sentence_31

The group Sn is the semidirect product of An and any subgroup generated by a single transposition. Symmetric group_sentence_32

Furthermore, every permutation can be written as a product of adjacent transpositions, that is, transpositions of the form (a a+1). Symmetric group_sentence_33

For instance, the permutation g from above can also be written as g = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5). Symmetric group_sentence_34

The sorting algorithm bubble sort is an application of this fact. Symmetric group_sentence_35

The representation of a permutation as a product of adjacent transpositions is also not unique. Symmetric group_sentence_36

Cycles Symmetric group_section_6

A cycle of length k is a permutation f for which there exists an element x in {1, ..., n} such that x, f(x), f(x), ..., f(x) = x are the only elements moved by f; it is required that k ≥ 2 since with k = 1 the element x itself would not be moved either. Symmetric group_sentence_37

The permutation h defined by Symmetric group_sentence_38

is a cycle of length three, since h(1) = 4, h(4) = 3 and h(3) = 1, leaving 2 and 5 untouched. Symmetric group_sentence_39

We denote such a cycle by (1 4 3), but it could equally well be written (4 3 1) or (3 1 4) by starting at a different point. Symmetric group_sentence_40

The order of a cycle is equal to its length. Symmetric group_sentence_41

Cycles of length two are transpositions. Symmetric group_sentence_42

Two cycles are disjoint if they move disjoint subsets of elements. Symmetric group_sentence_43

Disjoint cycles commute: for example, in S6 there is the equality (4 1 3)(2 5 6) = (2 5 6)(4 1 3). Symmetric group_sentence_44

Every element of Sn can be written as a product of disjoint cycles; this representation is unique up to the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point. Symmetric group_sentence_45

Special elements Symmetric group_section_7

Certain elements of the symmetric group of {1, 2, ..., n} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set). Symmetric group_sentence_46

The order reversing permutation is the one given by: Symmetric group_sentence_47

This is the unique maximal element with respect to the Bruhat order and the longest element in the symmetric group with respect to generating set consisting of the adjacent transpositions (i i+1), 1 ≤ i ≤ n − 1. Symmetric group_sentence_48

so it thus has sign: Symmetric group_sentence_49

which is 4-periodic in n. Symmetric group_sentence_50

Note that the reverse on n elements and perfect shuffle on 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic. Symmetric group_sentence_51

Conjugacy classes Symmetric group_section_8

The conjugacy classes of Sn correspond to the cycle structures of permutations; that is, two elements of Sn are conjugate in Sn if and only if they consist of the same number of disjoint cycles of the same lengths. Symmetric group_sentence_52

For instance, in S5, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. Symmetric group_sentence_53

A conjugating element of Sn can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Symmetric group_sentence_54

Continuing the previous example: Symmetric group_sentence_55

which can be written as the product of cycles, namely: (2 4). Symmetric group_sentence_56

This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is, Symmetric group_sentence_57

It is clear that such a permutation is not unique. Symmetric group_sentence_58

Low degree groups Symmetric group_section_9

See also: Representation theory of the symmetric group § Special cases Symmetric group_sentence_59

The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately. Symmetric group_sentence_60

Symmetric group_description_list_1

  • S0 and S1: The symmetric groups on the empty set and the singleton set are trivial, which corresponds to 0! = 1! = 1. In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S0, its only member is the empty function.Symmetric group_item_1_4

Symmetric group_description_list_2

  • S2: This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group and is thus abelian. In Galois theory, this corresponds to the fact that the quadratic formula gives a direct solution to the general quadratic polynomial after extracting only a single root. In invariant theory, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting fs(x, y) = f(x, y) + f(y, x), and fa(x, y) = f(x, y) − f(y, x), one gets that 2⋅f = fs + fa. This process is known as symmetrization.Symmetric group_item_2_5

Symmetric group_description_list_3

  • S3: S3 is the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from S3 to S2 corresponds to the resolving quadratic for a cubic polynomial, as discovered by Gerolamo Cardano, while the A3 kernel corresponds to the use of the discrete Fourier transform of order 3 in the solution, in the form of Lagrange resolvents.Symmetric group_item_3_6

Symmetric group_description_list_4

  • S4: The group is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, 9, 8 and 6 permutations, of the cube. Beyond the group A4, S4 has a Klein four-group V as a proper normal subgroup, namely the even transpositions {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, with quotient S3. In Galois theory, this map corresponds to the resolving cubic to a quartic polynomial, which allows the quartic to be solved by radicals, as established by Lodovico Ferrari. The Klein group can be understood in terms of the Lagrange resolvents of the quartic. The map from S4 to S3 also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree n of dimension below n − 1, which only occurs for n = 4.Symmetric group_item_4_7

Symmetric group_description_list_5

  • S5: S5 is the first non-solvable symmetric group. Along with the special linear group SL(2, 5) and the icosahedral group A5 × S2, S5 is one of the three non-solvable groups of order 120, up to isomorphism. S5 is the Galois group of the general quintic equation, and the fact that S5 is not a solvable group translates into the non-existence of a general formula to solve quintic polynomials by radicals. There is an exotic inclusion map S5 → S6 as a transitive subgroup; the obvious inclusion map Sn → Sn+1 fixes a point and thus is not transitive. This yields the outer automorphism of S6, discussed below, and corresponds to the resolvent sextic of a quintic.Symmetric group_item_5_8

Symmetric group_description_list_6

  • S6: Unlike all other symmetric groups, S6, has an outer automorphism. Using the language of Galois theory, this can also be understood in terms of Lagrange resolvents. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map S5 → S6 as a transitive subgroup (the obvious inclusion map Sn → Sn+1 fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S6—see automorphisms of the symmetric and alternating groups for details.Symmetric group_item_6_9

Symmetric group_description_list_7

  • Note that while A6 and A7 have an exceptional Schur multiplier (a triple cover) and that these extend to triple covers of S6 and S7, these do not correspond to exceptional Schur multipliers of the symmetric group.Symmetric group_item_7_10

Maps between symmetric groups Symmetric group_section_10

Other than the trivial map Sn → C1 ≅ S0 ≅ S1 and the sign map Sn → S2, the most notable homomorphisms between symmetric groups, in order of relative dimension, are: Symmetric group_sentence_61

Symmetric group_unordered_list_8

  • S4 → S3 corresponding to the exceptional normal subgroup V < A4 < S4;Symmetric group_item_8_11
  • S6 → S6 (or rather, a class of such maps up to inner automorphism) corresponding to the outer automorphism of S6.Symmetric group_item_8_12
  • S5 → S6 as a transitive subgroup, yielding the outer automorphism of S6 as discussed above.Symmetric group_item_8_13

There are also a host of other homomorphisms Sm → Sn where n > m. Symmetric group_sentence_62

Relation with alternating group Symmetric group_section_11

For n ≥ 5, the alternating group An is simple, and the induced quotient is the sign map: An → Sn → S2 which is split by taking a transposition of two elements. Symmetric group_sentence_63

Thus Sn is the semidirect product An ⋊ S2, and has no other proper normal subgroups, as they would intersect An in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in An (and thus themselves be An or Sn). Symmetric group_sentence_64

Sn acts on its subgroup An by conjugation, and for n ≠ 6, Sn is the full automorphism group of An: Aut(An) ≅ Sn. Symmetric group_sentence_65

Conjugation by even elements are inner automorphisms of An while the outer automorphism of An of order 2 corresponds to conjugation by an odd element. Symmetric group_sentence_66

For n = 6, there is an exceptional outer automorphism of An so Sn is not the full automorphism group of An. Symmetric group_sentence_67

Conversely, for n ≠ 6, Sn has no outer automorphisms, and for n ≠ 2 it has no center, so for n ≠ 2, 6 it is a complete group, as discussed in automorphism group, below. Symmetric group_sentence_68

For n ≥ 5, Sn is an almost simple group, as it lies between the simple group An and its group of automorphisms. Symmetric group_sentence_69

Sn can be embedded into An+2 by appending the transposition (n + 1, n + 2) to all odd permutations, while embedding into An+1 is impossible for n > 1. Symmetric group_sentence_70

Generators and relations Symmetric group_section_12

where 1 represents the identity permutation. Symmetric group_sentence_71

This representation endows the symmetric group with the structure of a Coxeter group (and so also a reflection group). Symmetric group_sentence_72

Other possible generating sets include the set of transpositions that swap 1 and i for 2 ≤ i ≤ n, and a set containing any n-cycle and a 2-cycle of adjacent elements in the n-cycle. Symmetric group_sentence_73

Subgroup structure Symmetric group_section_13

A subgroup of a symmetric group is called a permutation group. Symmetric group_sentence_74

Normal subgroups Symmetric group_section_14

The normal subgroups of the finite symmetric groups are well understood. Symmetric group_sentence_75

If n ≤ 2, Sn has at most 2 elements, and so has no nontrivial proper subgroups. Symmetric group_sentence_76

The alternating group of degree n is always a normal subgroup, a proper one for n ≥ 2 and nontrivial for n ≥ 3; for n ≥ 3 it is in fact the only non-identity proper normal subgroup of Sn, except when n = 4 where there is one additional such normal subgroup, which is isomorphic to the Klein four group. Symmetric group_sentence_77

The symmetric group on an infinite set does not have a subgroup of index 2, as Vitali (1915) proved that each permutation can be written as a product of three squares. Symmetric group_sentence_78

However it contains the normal subgroup S of permutations that fix all but finitely many elements, which is generated by transpositions. Symmetric group_sentence_79

Those elements of S that are products of an even number of transpositions form a subgroup of index 2 in S, called the alternating subgroup A. Symmetric group_sentence_80

Since A is even a characteristic subgroup of S, it is also a normal subgroup of the full symmetric group of the infinite set. Symmetric group_sentence_81

The groups A and S are the only non-identity proper normal subgroups of the symmetric group on a countably infinite set. Symmetric group_sentence_82

This was first proved by Onofri (1929) and independently Schreier-Ulam (1934). Symmetric group_sentence_83

For more details see (, Ch. Symmetric group_sentence_84

11.3) or (, Ch. Symmetric group_sentence_85

8.1). Symmetric group_sentence_86

Maximal subgroups Symmetric group_section_15

The maximal subgroups of the finite symmetric groups fall into three classes: the intransitive, the imprimitive, and the primitive. Symmetric group_sentence_87

The intransitive maximal subgroups are exactly those of the form Sym(k) × Sym(n − k) for 1 ≤ k < n/2. Symmetric group_sentence_88

The imprimitive maximal subgroups are exactly those of the form Sym(k) wr Sym(n/k) where 2 ≤ k ≤ n/2 is a proper divisor of n and "wr" denotes the wreath product acting imprimitively. Symmetric group_sentence_89

The primitive maximal subgroups are more difficult to identify, but with the assistance of the O'Nan–Scott theorem and the classification of finite simple groups, () gave a fairly satisfactory description of the maximal subgroups of this type according to (, p. 268). Symmetric group_sentence_90

Sylow subgroups Symmetric group_section_16

The Sylow subgroups of the symmetric groups are important examples of p-groups. Symmetric group_sentence_91

They are more easily described in special cases first: Symmetric group_sentence_92

The Sylow p-subgroups of the symmetric group of degree p are just the cyclic subgroups generated by p-cycles. Symmetric group_sentence_93

There are (p − 1)!/(p − 1) = (p − 2)! Symmetric group_sentence_94

such subgroups simply by counting generators. Symmetric group_sentence_95

The normalizer therefore has order p·(p − 1) and is known as a Frobenius group Fp(p−1) (especially for p = 5), and is the affine general linear group, AGL(1, p). Symmetric group_sentence_96

The Sylow p-subgroups of the symmetric group of degree p are the wreath product of two cyclic groups of order p. For instance, when p = 3, a Sylow 3-subgroup of Sym(9) is generated by a = (1 4 7)(2 5 8)(3 6 9) and the elements x = (1 2 3), y = (4 5 6), z = (7 8 9), and every element of the Sylow 3-subgroup has the form axyz for 0 ≤ i,j,k,l ≤ 2. Symmetric group_sentence_97

The Sylow p-subgroups of the symmetric group of degree p are sometimes denoted Wp(n), and using this notation one has that Wp(n + 1) is the wreath product of Wp(n) and Wp(1). Symmetric group_sentence_98

In general, the Sylow p-subgroups of the symmetric group of degree n are a direct product of ai copies of Wp(i), where 0 ≤ ai ≤ p − 1 and n = a0 + p·a1 + ... + p·ak (the base p expansion of n). Symmetric group_sentence_99

For instance, W2(1) = C2 and W2(2) = D8, the dihedral group of order 8, and so a Sylow 2-subgroup of the symmetric group of degree 7 is generated by { (1,3)(2,4), (1,2), (3,4), (5,6) } and is isomorphic to D8 × C2. Symmetric group_sentence_100

These calculations are attributed to () and described in more detail in (, p. 176) harv error: no target: CITEREFRotman1995 (help). Symmetric group_sentence_101

Note however that (, p. 26) attributes the result to an 1844 work of Cauchy, and mentions that it is even covered in textbook form in (, §39–40). Symmetric group_sentence_102

Transitive subgroups Symmetric group_section_17

A transitive subgroup of Sn is a subgroup whose action on {1, 2, ,..., n} is transitive. Symmetric group_sentence_103

For example, the Galois group of a (finite) Galois extension is a transitive subgroup of Sn, for some n. Symmetric group_sentence_104

Cayley's theorem Symmetric group_section_18

Cayley's theorem states that every group G is isomorphic to a subgroup of some symmetric group. Symmetric group_sentence_105

In particular, one may take a subgroup of the symmetric group on the elements of G, since every group acts on itself faithfully by (left or right) multiplication. Symmetric group_sentence_106

Automorphism group Symmetric group_section_19

Further information: Automorphisms of the symmetric and alternating groups Symmetric group_sentence_107

Symmetric group_table_general_0

nSymmetric group_cell_0_0_0 Aut(Sn)Symmetric group_cell_0_0_1 Out(Sn)Symmetric group_cell_0_0_2 Z(Sn)Symmetric group_cell_0_0_3
n ≠ 2, 6Symmetric group_cell_0_1_0 SnSymmetric group_cell_0_1_1 C1Symmetric group_cell_0_1_2 C1Symmetric group_cell_0_1_3
n = 2Symmetric group_cell_0_2_0 C1Symmetric group_cell_0_2_1 C1Symmetric group_cell_0_2_2 S2Symmetric group_cell_0_2_3
n = 6Symmetric group_cell_0_3_0 S6 ⋊ C2Symmetric group_cell_0_3_1 C2Symmetric group_cell_0_3_2 C1Symmetric group_cell_0_3_3

For n ≠ 2, 6, Sn is a complete group: its center and outer automorphism group are both trivial. Symmetric group_sentence_108

For n = 2, the automorphism group is trivial, but S2 is not trivial: it is isomorphic to C2, which is abelian, and hence the center is the whole group. Symmetric group_sentence_109

For n = 6, it has an outer automorphism of order 2: Out(S6) = C2, and the automorphism group is a semidirect product Aut(S6) = S6 ⋊ C2. Symmetric group_sentence_110

In fact, for any set X of cardinality other than 6, every automorphism of the symmetric group on X is inner, a result first due to () harv error: no target: CITEREFSchreierUlam1937 (help) according to (, p. 259). Symmetric group_sentence_111

Homology Symmetric group_section_20

See also: Alternating group § Group homology Symmetric group_sentence_112

The group homology of Sn is quite regular and stabilizes: the first homology (concretely, the abelianization) is: Symmetric group_sentence_113

The first homology group is the abelianization, and corresponds to the sign map Sn → S2 which is the abelianization for n ≥ 2; for n < 2 the symmetric group is trivial. Symmetric group_sentence_114

This homology is easily computed as follows: Sn is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps Sn → Cp are to S2 and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Symmetric group_sentence_115

Thus the only possible maps Sn → S2 ≅ {±1} send an involution to 1 (the trivial map) or to −1 (the sign map). Symmetric group_sentence_116

One must also show that the sign map is well-defined, but assuming that, this gives the first homology of Sn. Symmetric group_sentence_117

The second homology (concretely, the Schur multiplier) is: Symmetric group_sentence_118

This was computed in (), and corresponds to the double cover of the symmetric group, 2 · Sn. Symmetric group_sentence_119

The homology "stabilizes" in the sense of stable homotopy theory: there is an inclusion map Sn → Sn+1, and for fixed k, the induced map on homology Hk(Sn) → Hk(Sn+1) is an isomorphism for sufficiently high n. This is analogous to the homology of families Lie groups stabilizing. Symmetric group_sentence_120

The homology of the infinite symmetric group is computed in (), with the cohomology algebra forming a Hopf algebra. Symmetric group_sentence_121

Representation theory Symmetric group_section_21

Main article: Representation theory of the symmetric group Symmetric group_sentence_122

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. Symmetric group_sentence_123

This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles. Symmetric group_sentence_124

The symmetric group Sn has order n!. Symmetric group_sentence_125

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 representation by the same set that parametrizes conjugacy classes, namely by partitions of n or equivalently Young diagrams of size n. Symmetric group_sentence_126

Each such irreducible representation can 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. Symmetric group_sentence_127

Over other fields the situation can become much more complicated. Symmetric group_sentence_128

If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KSn is semisimple. Symmetric group_sentence_129

In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary). Symmetric group_sentence_130

However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. Symmetric group_sentence_131

In this context it is more usual to use the language of modules rather than representations. Symmetric group_sentence_132

The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. Symmetric group_sentence_133

The modules so constructed are called Specht modules, and every irreducible does arise inside some such module. Symmetric group_sentence_134

There are now fewer irreducibles, and although they can be classified they are very poorly understood. Symmetric group_sentence_135

For example, even their dimensions are not known in general. Symmetric group_sentence_136

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. Symmetric group_sentence_137

See also Symmetric group_section_22

Symmetric group_unordered_list_9

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: group.