5. Representations of the Symmetric Groups S n

5. Representations of the Symmetric Groups Sn 5.1 One-Dimensional Representations 5.2 Partitions and Young Diagrams 5.3 Symmetrizers and Anti-Symmetrizers of Young Tableaux 5.4 Irreducible Representations of Sn 5.5 Symmetry Classes of Tensors App III & IV

Importance of Sn: • Cayley's theorem: Every group of order n is isomorphic to a subgroup of Sn • Construction of IRs of classical groups, e.g., GL(n), U(n), SU(n), etc • Identical particles Basic tools: Young diagrams, Young tableaux Symmetrizers, anti-symmetrizers, Irreducible symmetrizers ( Idempotents / Projections ) nth rank tensors on m-D space : Sn + GL(m)

5.1. 1-D Reps Every Sn has a non-trivial invariant subgroup, the alternating group, An = { All even permutations }  Sn / An C2 • Every Sn has two 1-D reps: Identity rep 1 : D(p) = 1  p 2 : D(p) = (–) p

Theorem 5.1: Symmetrizer & antisymmetrizer are essentially idempotent & primitive Proof:   s is essentially idempotent & primitive   a is essentially idempotent & primitive

 s & a generate inequivalent IRs of Sn on Sn Basis vectors: Since 

5.2. Partitions and Young Diagrams Definition 5.1: Partition of n , Young Diagram A partition  of integer n is an ordered sequence of integers such that & 2 partitions  &  are equal if positive negative if the 1st non-zero member in sequence is A partition is represented graphically by a Young diagram of n squares arranged in r rows, the jth of which contains j squares

Example 1: n = 3 3 distinct partitions: { 3 } : { 2, 1 } { 1, 1, 1 } Example 2: n = 4 5 distinct partitions: { 4 } : { 3, 1 } { 2, 2 } { 2, 1, 1 } { 1, 1, 1,1 }

Partition of n ~ Classes of Sn Let there be j j-cycles in an element in a class of Sn  Then ( most k = 0 )  is a partition of n Theorem 5.2: Number of distinct Young diagram for n = Number of classes in Sn = Number of inequivalent IRs in Sn

Example: S3 = { e, (123), (132) , (23), (13), (12) } e = (1)(2)(3) (12) = (12)(3)

Definition 5.2: Young Tableau, Normal Tableau, Standard Tableau • A tableau is a diagram filled with a distinct number (1,…,n) in each square. • Young Tableau: Numbers filled with no particular order • Normal Tableau : Numbers filled consecutively from left to right & top to bottom • Standard Tableau p : Numbers ordered from left to right in each row & top to bottom in each column Example: S4 Standard tableaux p Normal tableaux  p = (3,4) p = (2,3)

5.3. Symmetrizers & Anti-Symmetrizers of Young Tableaux Definition 5.3: Horizontal and Vertical Permutations Let p be any tableau. A horizontal ( vertical ) permutation hp ( vp ) is a permutation that does not exchange numbers between different rows (columns). Each cycle in hp ( vp ) must contain numbers that appear in the same row (column). Definition 5.4: Symmetrizer, Anti-symmetrizer, Irreducible Symmetrizer Symmetrizer: Anti-symmetrizer: Irreducible symmetrizer:

Example: S3 • Observations: • { hp } and { vp } are each a subgroup of S3. • s and a are total (anti-)symmetrizers of these subgroups. Also: ( Rearrangement theorem used on subgroup { h} )  s and a are essentially idempotent. • eare primitive idempotents : • Cases e1 & e3 are obvious. For case e2, see Problem 5.3.

{ e } generates a set of inequivalent IRs of S3. • Cases e1 & e3 are obvious. • Case e2 is proved by showing that • { p e2  p  S3 } spans a 2–D subspace (left ideal) of S3. QED

Similarly e2(23) also generate a 2-D IR but it is equivalent to that from e2. • The left ideal is however distinct from that of e2. It is spanned by & • The group algebra is a direct sum of the 4 left ideals generated by the standard tableaux e1 , e2 , e2(23) and e3 . • The identity can be decomposed as  DR is fully reduced by the e 's of the standard tableaux

Summary of the lemmas proved in Appendix IV: Lemma IV.1: xp = p x p–1 Lemma IV.2: For a given tableau  { h } & { v } are each a subgroup of Sn. Lemma IV.3: Given  and p  Sn.  at least 2 numbers in one row of  which appear in the same column of  p.  Lemma IV.4: Given  and p  Sn  ( ~ denotes transpositions )

Lemma IV.5: Given  and r  G.  of Lemma IV.6: Given 2 distinct diagrams  > , Lemma IV.7: The linear group transformations on Vmn , spans the space K of all symmetry-preserving linear transformations.

5.4. Irreducible Representations of Sn Theorem 5.3: ( Superscripts p in p, sp…, are omitted ) For a given Young tableau  : and ( e is essentially idempotent ) Proof: By lemma IV.2 : By lemma IV.5 : where of where of Since e  s and e  a while (–)e = 1, we have αe  0 QED

Theorem 5.4: e  IR e is a primitive idempotent. It generates an IR of Sn on Sn. Proof: ( Theorem 5.3 ) By theorem III.3, e is a primitive idempotent. QED Theorem 5.5: Equivalent IRs IRs generated by e and ep , p  Sn, are equivalent Proof: ( Theorem 5.3 ) Proof is completed by theorem III.4.

Theorem 5.6: Inequivalent IRs e & e generate inequivalent IRs if    ( different Young diagrams ) Proof: Let  >  & p  Sn . Lemma IV.6  QED by theorem III.4 Example: e1, e2, e3 of S3 generate inequivalent IRs Corollary:  Proof: Case  >  is proved in lemma IV.6. Case  <  is left as exercise.

Theorem 5.7: IRs of Sn { e } of all normal tableaux generate all inequivalent IRs of Sn. • Proof: • Number of inequivalent IRs = Number of Young diagrams ( theorem 5.2 ) • Each normal diagram begets an e • Every e generates an inequivalent IR ( theorem 5.6 ) • Theorem 5.8: Decomposition of DR • Left ideals Lagenerated by e a 's associated with distinct standard tableaux are linearly independent. 2. Proof: See W. Miller, "Symmetry Groups & Their Applications", Academic Press (1972)

5.5. Symmetry Classes of Tensors Let Vm be an m-D vector space with basis { | i  , i = 1, …, m } { g } be all invertible linear transformations on Vm  { g } = GL(m, C) = General linear group = Gm Natural m-D rep of Gm on Vm : g i j = ( i, j ) element of an invertible m m matrix Definition 5.5: Tensor Space Vmn

Natural basis of Vmn : = tensor components of x Natural (nm)-D rep of Gm on Vmn :

Action of Sn on Vmn : with Natural n-D rep of Sn on Vmn :

Coming Attractions: • D[Gm] & D[Sn] are in general reducible. • D[Sn] can be decomposed using e 's of Sn. • Since p & g commutes, Gm can be decomposed using e's of Sn from reduction of Vmn.

Lemma 5.1: Gm &Sn are symmetry preserving onVmn For both D[Gm] & D[Sn] : where Proof: Follows directly from the explicit form of D[Gm] & D[Sn]. Theorem 5.9: p g = g p  g Gm & p Sn QED

Example 1: V22 Basis of V22 : S2 = { e, (12) }

Preview: In the decomposition of Vmn using ep, one gets 1. An irreducible invariant subspace wrt Sn: 2. An irreducible invariant subspace wrt Gm: • The decomposed Vmn has basis | , , a , where •  denotes a symmetry class / Young diagram, •  labels the irreducible invariant subspaces under Sn, and • a labels the irreducible invariant subspaces under Gm

Definition 5.6: Tensors of Symmetry p & Tensors of Symmetry Class  Given Young tableau p : Tensors of Symmetry p = Given Young diagram labelled : Tensors of Symmetry Class  =

Theorem 5.10: Let ( α fixed ) • T(α) is an irreducible invariant subspace wrt Sn. • T(α)   realization of Sn on T(α) coincides • with IRs generated by e on Sn. Proof of 1:  For some  QED Proof of 2:  Let { ri e } be a basis of L, then { ri e |   } is a basis for T().  If on Sn then on T() QED

= s = total symmetrizer    Ls is 1-D   Ts( ) = totally symmetric tensors  Realization of Sn on Ts( ) is the identity representation D1(p) = 1  p

Example 2: V23 ( Symmetry class s ) 4 distinct totally symmetric tensors (  = s ) can be generated: Ts(1) 1.  2.  Ts(2) 3.  Ts(3) 4.  Ts(4) Each Ts() is invariant under S3 All 4 | s, j, 1  together span a subspace Ts invariant under G2 There is no symmetry class a for V23 Problem 5.6: Symmetry class a exists only in Vmn with m n.

Example 3: Totally anti-symmetric tensor   1 & only 1 totally anti-symmetric tensor  in Vnn. n = 2: if ( i j k ) is permutations of (123) n = 3: Example 4: Vm2 , m  2 m ( m+1) / 2 distinct totally symmetric tensors: m ( m–1) / 2 distinct totally anti-symmetric tensors:

Example 5: V23 mixed symmetry  = m 2 independent irreducible invariant subspaces of tensors with mixed symmetry can be generated. Normal tableau Standard tableau 1. | m,1,1  & | m,1,2  span the 2-D subspace Tm(1), invariant under S3

2. | m,2,1  & | m,2,2  span the 2-D subspace Tm(2), invariant under S3  Tm(1) is invariant under G2. Ditto

Summary: The 8-D V23 is decomposed into 4 1-D class s & 2 2-D class m irreducible subspaces invariant under S3 Basis | , , *  for each T() is obtained by applying all standard tableaux p to a single |   The 8-D V23 is decomposed into 1 4-D class s & 2 2-D class m irreducible subspaces invariant under G2 Basis | , *, p  for each T (p) is obtained by applying each standard tableaux p to all |  

T() = span { ep |   } is invariant under Sn Theorem 5.11: 1. Either or ( disjoint ) 2. if    (different symmetries ) Proof of 1: Either or T() & T() has at least 1 non-zero element in common, i.e.,    Proof of 2: is also invariant under Sn. Since T() & T() are irreducible &    QED

Observations: Theorem 5.11 implies Each T() is invariant under Sn Basis | , , *  for each T() is obtained by applying ep of all standard tableaux p to a single |   It is permissible to the same D(Sn) for all 's : a,b = 1, …, dim T() dim T() = Number of standard tableaux * of symmetry 

Theorem 5.12: is invariant under Gm IR of Gm on T (a) satisfies Reminder: | , *, a  is obtained by applying ea of standard a to all |   Proof: Since theorem 5.9     QED Schur's lemma:

Theorem 5.13: IRs of Gm Reps of Gm on T(a) of Vmn are IRs. Reminder: | , *, a  is obtained by applying ea of standard a to all |   Outline of Proof: For complete proof, see W. Miller, "Symmetry Groups & Their Applications", Academic Press (72) Rationale: Since Gm is the largest group that commutes with Sn on T(a) of Vmn, the operators D(g) should be complete & hence irreducible. Let A be a linear operator on T(a) : Since x & y belong to the same symmetry class , A must be symmetry preserving, i.e., Lemma 5.1 states that gGm on Vmn are symmetry preserving. Lemma IV.7  A is a linear combination of D(g).  D(g) is irreducible

5. Representations of the Symmetric Groups S n