Decomposition numbers for symmetric groups Karin Erdmann University of Oxford, UK erdmann@maths.ox.ac.uk FPSAC, July 2009
Representations of symmetric groups G = S n , = finite-dimensional K -vector space. V Representation = group homomorphism ρ : G → GL ( V ). = G -module [ vg := ( v )( gρ ), g ∈ G, v ∈ V .] V 1
Ω { 2 } = 2-element subsets of { 1 , 2 , . . . , n } . K = Z 2 . K Ω { 2 } = M ( n − 2 , 2) permutation module of S n . { i, j } g = { ( i ) g, ( j ) g } Same for M ( n − 3 , 3) ? Composition factors? Qu. Same for eg M ( n − 5 , 3 , 2) ? Qu. 2
Specht modules S λ := Specht module. λ partition of n , • characteristic-free. • explicit: submodule of permutation module. S ( n ) = the trivial module. Eg K Ω = Span { v i } ∼ = M ( n − 1 , 1) . Ω = { 1 , 2 , . . . , n } , S ( n − 1 , 1) ∼ � � = { c i v i : c i = 0 } ⊂ K Ω . i i 3
S λ is simple. χ λ = the character of S λ • K = C : • char( K ) = p > 0: S µ has a unique simple quotient D µ . If µ is p-regular, β µ = the Brauer character of D µ . [ β µ ( g ) = tr D µ ( g ) if g ∈ S n is p-regular]. µ is p-regular if it does not have p equal parts: 6551 ⊢ 17 is 3-regular, but is not 2-regular. g is p-regular if p does not divide any cycle length of g . 4
Decomposition numbers d µ,λ := [ S µ : D λ ] = # D λ in a composition series of S µ , Decomposition number. On p-regular elements of S n , χ λ = d λ,µ β µ � µ D ( n ) = trivial module. EX � p | n 1 [ S ( n − 1 , 1) : D ( n ) ] = 0 else If p | n then χ ( n − 1 , 1) = β ( n − 1 , 1) + β ( n ) . 5
Decomposition matrix The decomposition matrix D = [ d µ,λ ] λ ⊢ n,µ ⊢ p n . • d λ,µ � = 0 ⇒ λ ≥ µ • d µ,µ = 1. D is upper uni-triangular. Some examples See Pictures. Find decomposition numbers!! Problem 6
Column removal p = 2 Example . . . = ( S (5 , 3) : D (6 , 2) ) = ( S (4 , 2) : D (5 , 1) ) = ( S (3 , 1) : D (4 , 0) ) = 1 Assume ˆ λ [ˆ µ ] is obtained from λ [ µ ] by removing the General first column. Theorem [G.D.James] If λ, µ have n non-zero parts and | λ | = | µ | then ( S λ : D µ ) = ( S ˆ λ : D ˆ µ ) . Similarly ’row removal’ & removal of ’blocks’, [S. Donkin]). 7
Proof (Column removal) The same holds for GL n . Prove this, then apply Schur functor. Write λ = λ n (1 n ) + ˆ GL n : λ , factorize the Schur polynomial: s λ = ( s (1 n ) ) λ n · s ˆ λ . Similarly for the formal characters of simple modules. Cancel the determinant part. 8
Two-part partitions • λ with r parts and d λ,µ � = 0 ⇒ µ has ≤ r parts. Theorem [G.D. James ’76] r = 2: � n − 2 j +1 � � 1 ≡ 1(mod p ) ( S ( n − k,k ) : D ( n − j,j ) ) = k − j 0 else [Column removal]: Get two quarter-infinite matrices which con- tain the decomposition matrices for all 2-part partitions. p = 2 and n even. See pcitures file. Example r ≥ 3 open. 9
Blocks If λ, µ are in different blocks, then d λ,µ = 0. Nakayama conjecture λ and γ are in the same p-block ⇔ λ, γ have the same p -core and the same p -weight; ⇔ λ, γ are in the same ’block’ of the decomposition matrix. Display partitions in B on an abacus with p runners, with ≥ pw beads. See the Pictures file. 10
Equivalences Suppose B = B κ,w is obtained from ¯ B = B ρ,w by swapping run- ners i, i + 1. • Assume # beads on runners i , i + 1 differ by ≥ w . [J. Scopes] Swapping runners induces Theorem (i) a bijection on partitions, (ii) preserves p − regularity and decomposition numbers. The block algebras B and ¯ B are Morita equivalent. For a fixed w , only finitely many blocks (up to Morita equiva- lence) as n varies. The first example in the pictures file satisfies the assumption. The second example does not. 11
The decomposition map Let R n := � λ ⊢ n Z χ λ , R n µ ⊢ p n Z β µ . Decomposition map: br := � ξ : R n → R n restrict to p-regular elements br , On p-regular elements, χ λ = � µ d λ,µ β µ . Recall Decomposition numbers: express the kernel of ξ w.r.to bases χ λ and β µ . Other descriptions of ker( ξ )? Question 12
Λ = ⊕ n ≥ 0 Λ n symmetric functions, characteristic isomorphism R := ⊕ n ≥ 0 R n char : Λ → char( s λ ) = χ λ M = GL n -module, M F = its Frobenius twist • ⇒ char( χ M F ) is in ker( ξ ). ψ p : Λ → Λ , x i → x p DEF: i , ring homomorphism. Then ψ p ( χ M ) = χ M F 13
Via char : Λ ∼ → R , get ring homomorphism ψ p : R → R. R n has Z -basis Theorem { ψ p ( χ λ ) · χ µ : µ p-regular } The subset of those with λ � = ∅ are a Z basis for ker( ξ ). Proof via symmetric functions. If χ γ occurs in ψ p ( χ λ ) · χ µ then γ ≥ λ p ∪ µ . And χ λ p ∪ µ occurs with multiplicity ± 1] δ = λ p ∪ µ . δ δ p -singular ⇒ ↔ row [ d δ, ∗ ] of D 14
p = 2, n = 4. EX ψ 2 ( χ (2) ) = χ (4) − χ (3 , 1) + χ (2 , 2) ψ 2 ( χ (1) ) · χ (2) = χ (4) + χ (2 , 2) − χ (2 , 1 2 ) ψ 2 ( χ (1 2 ) ) = χ (2 , 2) − χ (2 , 1 2 ) + χ (1 4 ) 15
Question at the beginning: M ( n − k,k ) has Specht filtration with Specht quotients S ( n ) , S ( n − 1 , 1) , S ( n − 2 , 2) , . . . , S ( n − k,k ) . Add corresponding rows of the decomposition matrix. [Depends on 2-adic expansion of n ] M ( n − 5 , 3 , 2) has Specht filtration, quotients from LR rule. De- composition numbers not known. 16
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