Hooks generate the representation ring of the symmetric group
HHOOKS GENERATE THE REPRESENTATION RING OF THESYMMETRIC GROUP
IVAN MARIN
Institut de Math´ematiques de JussieuUniversit´e Paris 7175 rue du ChevaleretF-75013 Paris
Abstract.
We prove that the representation ring of the symmetric group on n letters isgenerated by the exterior powers of its natural ( n − Keywords.
Symmetric group, representation rings.
MSC 2000.
Introduction
We let S n denote the symmetric group on n letters, and R ( S n ) its ordinary representationring, or equivalently the ring of its complex characters. It is a free Z -module with basis( V λ ) λ (cid:96) n of irreductible characters classically indexed by the set of partitions λ = [ λ , λ , . . . ]of n = λ + λ + . . . with λ ≥ λ ≥ · · · ≥
0. As usual, we identify such a λ with a Young (orFerrers) diagram, and we use the row-aligned, left-justified, top-to-bottom convention (e.g.the left-hand sides of figure 1 represent the partition [3 , , , size n of the partition λ is denoted | λ | .We refer to [FH] for classical facts about the correspondence between representations andpartitions. The notation we use here is such that the partition [ n ] is attached to the trivialrepresentation V [ n ] = , and the natural permutation representation S n < GL n ( C ) decom-poses itself as C n = + V with V = V [ n − , . Among the classical results that can be found in[FH] we recall that the exterior powers Λ k V for 0 ≤ k ≤ n provide irreducible representationsattached to the partitions [ n − k, k ]. Such representations or the corresponding partitionsare classically called hooks .The purpose of this note is to prove the following. Theorem 1.1.
For every n ≥ , the representation ring R ( S n ) is generated by the hooks Λ k V, ≤ k ≤ n − . Note that Λ k +1 C n = Λ k +1 V ⊕ Λ k V , hence the collection of the Λ k V and the collection ofthe Λ k C n span the same additive subgroup of R ( S n ). Another version of the same result isthus the following. Date : October 7, 2010. a r X i v : . [ m a t h . R T ] D ec IVAN MARIN
Theorem 1.2.
For every n ≥ , the representation ring R ( S n ) is generated by the represen-tations Λ k C n , ≤ k ≤ n − . This latter version can be compared with the similar classical result for GL n ( C ), that itsring of rational representations is generated by the Λ k C n (which, in terms of characters,simply means that the symmetric polynomials are generated by the elementary symmetricones – see e.g. [FH], (6.2) and appendix A).It has been communicated to us by J.-Y. Thibon that, when translated in the language ofsymmetric functions, the theorems above are equivalent to the results of Butler and Boorman(see [Bu, Boo] and also [STW]). The main point of this note is thus to show how to derivethis result from the strikingly simple formula of Dvir (see § A filtration on R ( S n )Let G be a finite group, V a faithful (finite-dimensional, complex, linear) representation of G and Irr( G ) the set of all irreducible representations. Then the representation ring R ( G ) isa free Z -module with basis Irr( G ), and each ρ ∈ Irr( G ) embeds into some V ⊗ r for r ∈ Z ≥ (Burnside-Molien, see e.g. [FH] problem 2.37). The level (or depth ) of ρ ∈ Irr( G ) with respectto V is defined to be N ( ρ ) = min { r ∈ Z ≥ | ρ (cid:44) → V ⊗ r } Obviously we have N ( ρ ⊗ ρ ) ≤ N ( ρ ) + N ( ρ ), N ( ) = 0. It follows that the subgroup F r of R ( G ) generated by the ρ ∈ Irr( G ) with N ( ρ ) ≤ r defines a ring filtration F ⊂ F ⊂ · · · ⊂ of R ( G ), hence a ring structure (gr R ( G ) , (cid:12) ) on the graded ringgr R ( G ) = + ∞ (cid:77) k =0 ( F k R ( G )) / ( F k − R ( G ))with the convention F − R ( G ) = { } . Notice that Irr( G ) provides a basis of R ( G ) as a Z -module.We now let G = S n . Considering S n − < S n through the natural embedding that leavesthe n -th letter untouched, we let Ind : R ( S n − ) → R ( S n ) and Res : R ( S n ) → R ( S n − )denote the usual induction and restriction morphisms.Recall that Res and Ind are easily described on Young diagrams by Young rule, as illustratedby figure 1. If λ is a Young diagram of size n , then Res V λ is the sum (without multiplicities) in R ( S n − ) of the V µ , with µ being deduced from λ by removing (respectively adding) one box.Similarly, if λ is a Young diagram of size n −
1, then Ind V λ is the sum (without multiplicities)in R ( S n ) of the V µ , with µ being deduced from λ by adding one box.The operator Ind Res on Young diagrams then means summing all V µ for µ a diagramdeduced from λ by moving one box, and δ ( λ ) copies of V λ where δ ( λ ) = { i | λ i (cid:54) = λ i +1 } (seefigure 2). OOKS GENERATE THE REPRESENTATION RING OF THE SYMMETRIC GROUP 3
IndRes
Figure 1.
Restriction and induction on Young diagrams
Figure 2.
Ind ◦ Res [2 , V = V [ n − , . By the above Young rule, we have C n = Ind = + V . Using theclassical formula U ⊗ Ind W (cid:39) Ind((Res U ) ⊗ W ) we get, for all U ∈ R ( S n ), U + U ⊗ V = U ⊗ ( + V ) = Ind Res U i.e. U ⊗ V = (Ind Res U ) − U . Because of this, N ( λ ) = N ( V λ ) can be determined combi-natorially. First note that, if V λ (cid:44) → V ⊗ ( r − ⊗ V , then V λ (cid:44) → V µ ⊗ V for some irreducible V µ (cid:44) → V ⊗ ( r − . An immediate consequence of the above remarks is thus that the number λ of boxes in the first row for λ satisfies λ ≥ µ −
1. By induction on r this yields r ≥ n − λ ,hence N ( λ ) ≥ n − λ . One then easily gets the following classical fact, for which we couldnot find an easy reference. Proposition 2.1.
For all λ (cid:96) n , we have N ( λ ) = n − λ .Proof. The proof is by induction on r = n − λ , the case r = 0 being clear. Let λ = [ λ , . . . , λ s ]with λ ≥ λ ≥ · · · ≥ λ s > n − λ = r +1. Since n − λ > s ≥
2. We consider µ (cid:96) n defined by µ = λ + 1, µ i = λ i for 1 < i < s , and µ s = λ s −
1. By the induction assumption, N ( µ ) = r and V µ (cid:44) → V ⊗ r . One of the components of V µ ⊗ V is V λ by the combinatorial rule,hence V λ (cid:44) → V µ ⊗ V (cid:44) → V ⊗ ( r +1) and the conclusion follows by induction. (cid:3) For a partition λ = [ λ , λ , . . . ] of n with λ i ≥ λ i +1 , we define the partition θ ( λ ) =[ λ , λ , . . . ] of n − λ . In diagrammatic terms, θ ( λ ) is the diagram deduced from λ by deletion IVAN MARIN
Figure 3.
The [2 , , , Figure 4. L [2 , , [2 , , [3 , , = 2 of the first row (see figure 5). Proposition 2.1 can thus be reformulated as | θ ( λ ) | = N ( λ ) . Dvir’s formula
For three partitions λ, µ, ν of arbitrary size, we let L λ,µ,ν denote the Littlewood-Richardsoncoefficient (see e.g. [FH]). A remarkable discovery of Y. Dvir is that the graded ring structure(gr R ( S n ) , (cid:12) ) is basically given by these coefficients.We first recall how to compute L λ,µ,ν with | ν | = | λ | + | µ | using the Littlewood-Richardsonrule : L λ,µ,ν is the number of ways λ , as a Young diagram, can be expanded into ν by usinga µ -expansion . Letting µ = [ µ , . . . , µ k ], such a µ -expansion is obtained by first adding µ boxes labelled by 1, then µ boxes labelled by 2, and so on (that is, at the r -th step we add µ r boxes labelled r to the preceedingly obtained diagram) so that(1) at each step, one still has a Young diagram(2) the labels strictly increase in each column(3) when the labels are listed from right to left in each row and starting with the top row,we have the following property. For each t ∈ [1 , | µ | ], the following holds : each label p occurs at least as many times as the label p + 1 (when it exists) in the first t entries.As an example, see figure 3 for the list of the [2 , , ,
1] and figure 4 for thetwo expansions leading to L [2 , , [2 , , [3 , , = 2. The reader can find in [FH] other examplesand further details on this combinatorics.For λ, µ, ν (cid:96) n , we let C λµν denote the structure constants V λ ⊗ V µ = (cid:80) ν C λµν V ν of R ( S n ).These constants, whose study has been initiated by Murnaghan (1938), are notoriously com-plicated to understand.For a partition λ = [ λ , λ , . . . ] with λ i ≥ λ i +1 , of n , define the partition θ ( λ ) = [ λ , λ , . . . ]of n − λ , and let d ( λ ) = | θ ( λ ) | = λ + λ + · · · = n − λ . By proposition 2.1 above we have OOKS GENERATE THE REPRESENTATION RING OF THE SYMMETRIC GROUP 5 l a a m o Figure 5. α = θ ( λ ), α ◦ and µ for λ = [5 , , , d ( λ ) = N ( λ ) = min { r ≥ | V λ (cid:44) → V ⊗ r } . In particular C λ,µ,ν = 0 whenever d ( ν ) > d ( λ )+ d ( µ ).Dvir’s formula can be stated as follows Theorem 3.1. (Dvir [D] , theorem 3.3) Let λ, µ, ν be partitions of n such that d ( λ ) + d ( µ ) = d ( ν ) . Then C λ,µ,ν = L θ ( λ ) ,θ ( µ ) ,θ ( ν ) . In particular we get, inside gr R ( S n ), the following formula : V λ (cid:12) V µ = (cid:88) d ( ν )= d ( λ )+ d ( µ ) L θ ( λ ) ,θ ( µ ) ,θ ( ν ) V ν . The proof
The main theorem is then an immediate consequence of the following proposition. For theproof of this proposition, we will associate to a Young diagram α = [ α , α , . . . ] its interior α ◦ defined by the partition α ◦ i = max(0 , α i − ∂α is defined to be theribbon made of the boxes in α which do not belong to α ◦ . The size | ∂α | of ∂α (that is, itsnumber of boxes) is clearly equal to the number of rows in α , or in other terms to the numberof nonzero parts of the partition α . Proposition 4.1.
The ring (gr R ( S n ) , (cid:12) ) is generated by the Λ k V , ≤ k ≤ n − .Proof. Recall that Λ k V = V [ n − k, k ] , and note that θ ([ n − k, k ]) = [1 k ]. In particular N (Λ k V ) = k . We identify each V λ with its image in gr R ( S n ) and let Q denote the sub-ring of gr R ( S n ) generated by the Λ k V . We prove that V λ ∈ Q for all partition λ of n ( λ (cid:96) n ), by induction on d ( λ ) = | θ ( λ ) | . We have d ( λ ) = 0 ⇒ λ = [ n ] ⇒ V λ = Λ V and d ( λ ) = 1 ⇒ λ = [ n − , ⇒ V λ = Λ V , hence V λ ∈ Q if d ( λ ) ≤
1. We thus assume d ( λ ) ≥ V µ ∈ Q for all partitions µ with d ( µ ) < d ( λ ).Letting α = θ ( λ ) we use another induction on | α ◦ | . Note that | α ◦ | ≤ | α | , with equalityonly if α = ∅ . More generally, the case | α ◦ | = 0 means that V λ = Λ | ∂α | V ∈ Q , so we canassume | α ◦ | ≥ r = | ∂α | = | α | − | α ◦ | . Since d ( λ ) ≥ θ ( λ ) (cid:54) = 0 and in particular r ≥ λ = n −| α | ≥ λ , hence n −| α ◦ | ≥ α ≥ α ◦ . We thus can introduce the partition µ =[ n − | α ◦ | , α ◦ , . . . , ] of n (see figure 5 for an example) and consider M = V µ (cid:12) Λ r V ∈ gr R ( S n ).Since | α ◦ | < | α | we have d ( µ ) < d ( λ ) hence V µ ∈ Q by the first induction assumption so M ∈ Q . Let ν (cid:96) n such that M has nonzero coefficient on V ν . We have d ( ν ) = d ( µ )+ r = d ( λ ),hence ν = n − | α | = λ , and this coefficient is L α ◦ , [1 r ] ,θ ( ν ) by Dvir formula. IVAN MARIN
By the Littlewood-Richardson rule, this coefficient L α ◦ , [1 r ] ,θ ( ν ) is the number of ways thatone can add boxes labelled 1 , . . . , r on the Young diagram of α ◦ with at most one box oneach row (with the graphic convention that α ◦ has α ◦ i boxes on the i -th row), the labelsincreasing along the rows, and such that the augmented diagram corresponds to θ ( ν ). Wethus clearly have L α ◦ , [1 r ] ,α = 1, this corresponding to adding a box marked i on the i -th rowfor each 1 ≤ i ≤ r . Moreover, if L α ◦ , [1 r ] ,θ ( ν ) is nonzero, then either θ ( ν ) has (strictly) morenonzero parts than α , which means that one box has been added to the empty ( r + 1)-st row,and in that case we know that V ν ∈ Q by the second induction hypothesis (as this means | ∂θ ( ν ) | > r = | ∂α | , hence | θ ( ν ) ◦ | < | α ◦ | since | α | = | θ ( ν ) | ); or, the r boxes have been addedto the first row, which implies θ ( ν ) = α hence ν = λ . We thus get M ≡ V λ modulo Q , V λ ∈ Q and the conclusion follows by induction. (cid:3) A careful look at the above proof shows that we proved a more technical but also moreprecise result. For λ, µ ∈ S n , we define λ ≺ µ if either N ( λ ) < N ( µ ), or N ( λ ) = N ( µ )and | θ ( λ ) ◦ | < | θ ( µ ) ◦ | , and we denote by R λ (resp. R λ ) the Z -submodule of R ( S n ) (resp.gr R ( S n )) spanned by the κ ∈ Irr( S n ) with κ ≺ λ . The above proof actually shows thefollowing. Proposition 4.2.
For every λ ∈ Irr( S n ) \ { } , there exists ˆ λ ∈ Irr( S n ) with ˆ λ ≺ λ and k ∈ Z ≥ such that ˆ λ (cid:12) Λ k V ∈ λ + R λ . Since F N ( κ ) − ( S n ) ⊂ R κ this immediately implies Corollary 4.3.
For every λ ∈ Irr( S n ) \ { } , there exists ˆ λ ∈ Irr( S n ) with ˆ λ ≺ λ and k ∈ Z ≥ such that ˆ λ ⊗ Λ k V ∈ λ + R λ . An application
One can use this result to give a proof of the well-known fact that all complex linearrepresentations of the symmetric group can actually be realized over Q . We first recall thefollowing lemma. Lemma 5.1.
Let G be a finite group, k a number field, ρ : G → GL N ( k ) a linear represen-tation of G defined over k , and ρ C : G → GL N ( C ) its complexification. If ϕ is an irreduciblesubrepresentation of ρ C occuring with multiplicity one whose character takes values in k , then ϕ can be realized over k .Proof. This is an immediate consequence of the fact that the projection on the ϕ -isotopiccomponent of ρ C is given by dim ϕ | G | (cid:80) g ∈ G χ ( g ) ρ ( g ) (see e.g. [FH] (2.32)), which is an endo-morphism of k N under our assumptions. (cid:3) We now can deduce the following well-known result.
Theorem 5.2.
Every complex linear representation of S n can be realized over Q .Proof. We use first that the natural permutation module C n is obviously realizable over Q ,and that C n = + V . This implies that the character associated to V is defined over Q ,hence V can be realized over Q by lemma 5.1 (or, directly, V can be identified to the rationalsubspace { ( x , . . . , x n ) ∈ Q n | x + · · · + x n = 0 } ). It follows that all the Λ r V can be realizedover Q .Since Irr( S n ) is clearly a well-founded set under ≺ , with minimal element , one can nowuse this relation to prove our statement by induction. OOKS GENERATE THE REPRESENTATION RING OF THE SYMMETRIC GROUP 7
Let λ ∈ Irr( S n ). Corollary 4.3 implies that there exists ˆ λ ≺ λ and k ∈ Z ≥ such that M = ˆ λ ⊗ Λ k , which is realizable over Q by our induction assumption, contains λ withmultiplicity 1, and has the property that the quotient representation M/λ is also realizableover Q by the same induction assumption. This proves that the character of λ takes valuesin Q , and then that λ is realizable over Q by lemma 5.1. This concludes the proof. (cid:3) Generalization attempts
The symmetric group is an irreducible complex (pseudo-)reflection group. Recall that sucha group is a finite subgroup W of GL( V ) for V some finite-dimensional complex vector spaceacted upon irreductibly by W , with W generated by its reflections, namely elements of GL( V )which fix a hyperplane. The dimension of V is called the rank of W .For such a group, it is a classical result of Steinberg that the representations Λ k V areirreducible (see e.g. [Bou] ch. 5 § hooks .Among other similarities, theorem 5.2 admits a natural generalization to these groups.Indeed, it can first be shown that the representation V can be realized over its character field k (i.e. the number field generated by the values taken by its character), sometimes called its field of definition . Moreover, it is a theorem of M. Benard that every representation of W can be realized over Q (see [Bena], and also [Bes], [MM] for other proofs), thus providing acomplete generalization of theorem 5.2. We now investigate to what extent theorem 1.1 couldbe generalized.The irreducible complex reflection groups have been classified by Shephard and Todd (see[ST]). There is an infinite series G ( de, e, r ) depending on three integral parameters d, e, r ,plus 34 exceptions G , . . . , G . For the representation theory of the G ( de, e, r ) we refer to[AR].Note that, for a given group with known character table, it is easy to check by computerwhether a given subset B of Irr( W ) generates R ( W ). Indeed, the ring R ( W ) = Z Irr( W ) is afree Z -module with basis Irr( W ) ; assume we are given a subset B ⊂
Irr( W ) with ∈ B , andlet A denote the subring of R ( W ) generated by B . The embedding R ( W ) ⊂ End Z R ( W ) (cid:39) End Z ( Z Irr( W )) identifies A with the minimal Z -submodule of Z Irr( W ) containing W whichis stable under multiplication by B . This identifies a ∈ A with a. ∈ Z Irr( W ). Starting withthe Z -module A = Z1 of rank 1, multiplication by the elements of B iteratively provides asequence of submodules A ⊂ A ⊂ . . . which eventually stops at A ∞ = A by noetherianityof the Z -module R ( W ).If W has rank 2, we are able to prove case-by-case the following. Proposition 6.1. If W is an irreducible complex reflection group of rank 2, then R ( W ) isgenerated by V and the 1-dimensional representations.Proof. The case of exceptional reflection groups is checked by computer, using the algorithmabove. The non-exceptional ones are the G ( de, e, W = G ( de, e, W have dimension at most 2. The ones of dimension 2 can beextended to G ( de, , e = 1. The group W is generated by t = diag(1 , ζ ) with ζ = exp(2i π/d ) and s the permutation matrix (1 2). Itstwo-dimensional representations are indexed by couples ( i, j ) with 0 ≤ i < j < d . We extendthis notation to i, j ∈ Z with j (cid:54)≡ i mod d by taking representatives modulo d and letting IVAN MARIN ( i, j ) = ( j, i ). A matrix model for the images of t and s in the representation ( r, r + k ) is t (cid:55)→ (cid:18) ζ r ζ r + k (cid:19) s (cid:55)→ (cid:18) (cid:19) In particular, V = (0 , , ⊗ (0 ,
1) is the sum of (0 ,
2) and 1-dimensional representations, and that (0 , ⊗ (0 , k ) =(0 , k + 1) + (1 , k ). Then we consider the 1-dimensional representation χ : t (cid:55)→ ζ, s (cid:55)→
1. Itis clear that ( i, j ) ⊗ χ = ( i + 1 , j + 1). Letting Q denote the subring of R ( W ) generated by V and the 1-dimensional representations, through tensoring by χ is it enough to show that(0 , k ) ∈ Q for all 1 ≤ k ≤ d . By definition (0 , ∈ Q , tensoring by (0 ,
1) yields (0 , ∈ Q ,and finally (0 , ⊗ (0 , k ) = (0 , k + 1) + χ ⊗ (0 , k −
1) proves the result by induction on k . (cid:3) Among the higher rank exceptional groups, we check by computer that the union of theΛ k V and the one-dimensional representations generates R ( W ) exactly for the groups G = H , G , G , G , G = H , G , G = E (but not E nor E !).In the more classical case of the Coxeter groups W of type B n and D n , it is easily checkedthat the subring generated by the Λ k V has not full rank in R ( W ) (for n ≥ U of dimension n − n − , , ∅ ) and { [ n − , , ∅} in the usual parametrizationsof their irreducible representations (see [GP]). These are deduced from V [ n − , ∈ Irr( S n )through a natural morphism W (cid:16) S n . A computer check for small values of n motivates thefollowing conjecture. Conjecture 6.2.
For W a Coxeter group of type B n or D n +1 , R ( W ) is generated by the Λ k V, Λ k U, k ≥ . The proof of such a conjecture would probably involve an understanding of the structureconstants in R ( W ) comparable to Dvir’s formula for S n . Unfortunately, the combinatorialstudy of the representation ring of these more general Coxeter groups seems to be only at thebeginning.For a group of type D n , it can be checked that the subring generated by such elements hassmaller rank already for D . This is a general phenomenon, as can be seen in the followingway. Recall that a group W of type D n is an index 2 subgroup of a Coxeter group (cid:102) W oftype B n . By Clifford theory, an irreducible representations of (cid:102) W parametrized by ( λ, µ ) with | λ | + | µ | = n restricts either to an irreducible representation { λ, µ } of W , precisely in the case λ (cid:54) = µ , or, in the case λ = µ , to a direct sum of two irreducibles usually denoted λ + and λ − .Note that such λ ± exist if and only if n is even.Choosing some s ∈ (cid:102) W \ W and letting Ad s : x (cid:55)→ sxs − be the automorphism of W inducedby s , the map ρ (cid:55)→ ρ ◦ Ad s induces a Z -linear involution η of R ( W ) which fixes the { λ, µ } and maps λ ± to λ ∓ . Letting R ( W ) η denote the invariant subspace, we have R ( W ) η = R ( W )if and only if n is odd. Clearly the Λ k V and Λ k U are always fixed by η , and this explainswhy the subring they generate cannot be R ( W ) when R ( W ) η (cid:54) = R ( W ). We do not have anyserious guess for a natural generating set in these cases. References [AR] S. Ariki,
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