Generalized Schur algebras 6 1 Robert May 0 Department of Mathematics and Computer Science 2 Longwood University n a 201 High Street, Farmville, Va. 23909 J rmay@longwood.edu 7 05/10/2014 ] A R . 1 Introduction h t a In [3] and [1] “generalizedSchur algebras”,B(n,r), were defined corresponding m to a family of monoids S which includes the full transformation semigroup r [ τr =Tr, the partial transformation semigroup τ¯r =PTr, and the rook monoid 1 ℜ ofsizer. Inmanycasesaparameterizationoftheirreduciblerepresentations r v of the algebra B(n,r) was obtained. In [2] these algebras were redefined as 1 the “right generalized Schur algebras” , B , corresponding to certain “double 1 R coset algebras”. Corresponding “left generalized Schur algebras” , B , were 7 L 1 also defined in [2]. In this paper we use the approach in [1] to study both 0 the right and left algebras. The algebras and their multiplication rules are . described in sections 2 through 4. In section 5, we obtain filtrations of these 1 0 algebras. These lead, in most cases, to parameterizations of the irreducible 6 representations for both B and B for a field of characteristic 0 (in section 6) R L 1 or positive characteristic p (in sections 7 and 8). : v i X 2 The semigroups S and the double coset alge- r r a bra A(S ,G) r Letτ¯ bethesetofallmapsα:{0,1,...,r}→{0,1,...,r} such that α(0)=0. r τ¯ is a semigroup under composition and is isomorphic to the partial transfor- r mationsemigroupPT . Letτ bethefulltransformationsemigroupandS the r r r symmetric groupon {1,2,...,r}. Any α in τ or S canbe extended to α¯ ∈τ¯ r r r by defining α¯(0) = 0, so we can regard S and τ as subsemigroups of τ¯ . We r r r will let S represent any subsemigroup of τ¯ which contains S . For example, r r r S couldbe the rook semigroup, ℜ = α∈τ¯ :∀i∈{1,2,...r}, α−1(i) 61 . r r r Our main examples for S will be S ,τ ,ℜ , and τ¯ . Note that S = τ ∩ r r(cid:8) r r r (cid:12) r (cid:12) r(cid:9) ℜ and τ¯ =ℜ ·τ . (cid:12) (cid:12) r r r r 1 EachS canbeidentifiedwithacertainsemigroupofmatrices. LetM (Z) r r+1 be the set of all (r+1)×(r+1) matrices with entries in Z (considered as a semigroup under matrix multiplication). For convenience, label the rows and columns for each m ∈ M (Z) from 0 to r. To each α ∈ S assign a matrix r+1 r 1 if i=α(j) m(α) ∈ M (Z) by setting m(α) = . Note that column j r+1 i,j 0 otherwise (cid:26) of m(α) contains exactly one non-zero entry, namely a 1 in row α(j). The 1 in column 0 is always in row 0. It is not hard to check that α 7→ m(α) gives an injective semigroup homomorphism S → M (Z), so we will identify S r r+1 r with its image in M (Z). For α ∈ S we will usually write just α for the r+1 r corresponding matrix m(α) . If α ∈ R , then m(α) has at most one 1 in rows r 1,2,...,r. If α ∈ τ , then m(α) has only one 1 in row 0 (in column 0). If r α∈S ,thenm(α)istheusualpermutationmatrixborderedwithanextrarow r 0 and column 0. LetΛ(r,n)bethesetofallcompositionsofrwithnparts. Soforλ∈Λ(r,n) n we have λ= {λ ∈Z:i=1,2,...,n}, λ > 0, λ =r. For each λ∈Λ(r,n) i i i i=1 define “λ-blocks” of integers, bλ, and a “YoungPsubgroup”, S ⊆ S ⊆ S as i λ r r follows. First, let bλ ={k ∈Z:0r. Then G = {id} and n (ν,β,µ) = 1 = n (λ,α,ν) for any λ,µ ∈ Λ(r,n) and ν R L α,β ∈S . Also r 1, if D =G αβG ; λ µ N(D(λ,α,ν),D(ν,β,µ),D)= (0, otherwise. Then substituting in the multiplication rule gives n (D(λ,αβ,µ)) R f(λ,α,ν)∗ f(ν,β,µ)= f(λ,αβ,µ) R n (D(λ,α,ν)) R For α=1 (the identity in S ), n (λ,1,ν)=1 giving r R f(λ,1,ν)∗ f(ν,β,µ)=n (D(λ,β,µ))f(λ,β,µ) R R If in addition λ=ν, then writing 1 =f(ν,1,ν) gives ν 1 ∗ f(ν,β,µ)=f(ν,β,µ). ν R There are similar rules for ∗ : L n (D(λ,αβ,µ)) L f(λ,α,ν)∗ f(ν,β,µ)= f(λ,αβ,µ) L n (D(λ,α,ν)) L f(λ,α,ν)∗ f(ν,1,µ)=n (D(λ,α,µ))f(λ,α,µ) L L f(λ,α,ν)∗ 1 =f(λ,α,ν). L ν Case 2: α=1 (the identity in S ) and G ⊆G . r ν λ Then for any ρ∈G , we have αρβ =ρβ ∈G βG , so ν λ µ o(G ), if D =G βG ; ν λ µ N(D(λ,1,ν),D(ν,β,µ),D)= (0, otherwise. Also G (λ,1,ν)=G =G (λ,1,ν), so R ν L n (D(λ,1,ν))=o(G )=n (D(λ,1,ν)). R ν L 5 Then substitution in the multiplication rule gives: n (D(λ,β,µ)) R f(λ,1,ν)∗ f(ν,β,µ)= f(λ,β,µ). R n (D(ν,β,µ)) R In particular, if λ=ν and 1 =f(ν,1,ν) then ν 1 ∗ f(ν,β,µ)=f(ν,β,µ). ν R Similarly, n (D(λ,β,µ)) L f(λ,1,ν)∗ f(ν,β,µ)= f(λ,β,µ) L n (D(ν,β,µ)) L 1 ∗ f(ν,β,µ)=f(ν,β,µ). ν L Case 3: β =1 and G ⊆G . ν µ Then for any ρ∈G , we have αρβ =αρ∈G αG , so ν λ µ o(G ), if D =G αG ; ν λ µ N(D(λ,α,ν),D(ν,1,µ),D)= (0 otherwise. Also G (ν,1,µ)=G =G (ν,1,µ), so R ν L n (D(ν,1,µ))=o(G )=n (D(ν,1,µ)). R ν L Then substitution in the multiplication rule gives: n (D(λ,α,µ)) R f(λ,α,ν)∗ f(ν,1,µ)= f(λ,α,µ) R n (D(λ,α,ν)) R In particular, if µ=ν and 1 =f(ν,1,ν) then ν f(λ,α,ν)∗ 1 =f(λ,α,ν). R ν Similarly, n (D(λ,α,µ)) L f(λ,α,ν)∗ f(ν,1,µ)= f(λ,α,µ) L n (D(λ,α,ν)) L f(λ,α,ν)∗ 1 =f(λ,α,ν). L ν Case 4: α=β =1, λ=µ, G ⊆G =G . ν λ µ As a special case of either case 2 or case 3, o(G ) λ f(λ,1,ν)∗ f(ν,1,λ)= f(λ,1,λ)=f(λ,1,ν)∗ f(ν,1,λ). R L o(G ) ν 6 In particular, when ν =λ=µ, 1 ∗ 1 =1 =1 ∗ 1 . ν R ν ν ν L ν So in both A and A , each1 =f(ν,1,ν) is anidempotent. In fact for either R L ν algebra, cases 2 and 3 can be used to check that {1 :ν ∈Λ(r,n)} is a family ν of orthogonal idempotents with ⊕ 1 =1 (the identity). ν ν∈Λ(r,n) 4 The left and right generalized Schur algebras Let R be a commutative domain with unit 1 and let ψ :Z→R be the natural ring homomophism such that ψ(1) = 1. Regard R as a right Z-module via ψ and form the tensor products LGSR = R⊗ZAL and RGSR = R⊗ZAR which we call the left and right generalized Schur algebras over R (for the monoid S ). (As in [2], for S =S , LGS is isomorphic to the classical Schur r r r R algebra over R, S (r,n), while RGS is isomorphic to the opposite algebra R R S (r,n)op.) The R algebras LGS , RGS are both free as R modules with R R R bases {1⊗f(λ,D,µ):λ,µ∈Λ(r,n),D ∈ M }. We will often denote a basis λ µ element 1⊗ f(λ,D,µ) by just f(λ,D,µ). Then all the multiplication rules given in section 3 apply if we identify each coefficient a ∈ Z occurring with its image ψ(a) ∈ R. As shown in [2], the isomorphism A ∼= C⊗AL ≡ LGSC is given by matching basis elements X(λ,D,µ) ↔ 1⊗f(λ,D,µ), where n (D) nL(D) L is the number of elements in any left G coset C ⊆ D ∈ M . Similarly, λ λ µ the isomorphism A ∼= C⊗AR ≡ RGSC is given by matching basis elements X(λ,D,µ) ↔1⊗f(λ,D,µ),wheren (D)isthe numberofelementsinanyright nR(D) R G coset C ⊆D ∈ M . µ λ µ 5 Filtrations of generalized Schur algebras and irreducible representations Let Λ=Λ(r,n) be the set of all compositions of r into n parts; Λ+ =Λ+(r)⊆ Λ, the set of all partitions of r. For λ ∈ Λ and 0 6 k 6 r, put L(λ,k) = #{i:λ =k} (= number of rows of length k in λ ). Define an equivalence i relation on Λ by λ ∼ λ ⇔ ∀i,L(λ ,i) = L(λ ,i). Notice that for each 1 2 1 2 λ ∈ Λ there is exactly one partition λ+ ∈ Λ+ with λ ∼ λ+. Let < be the “lexicographic” order on Λ+: For λ,µ ∈ Λ+, λ < µ ⇔ ∃k s.t. λ = µ for i < i i k while λ < µ . Then < gives a total ordering of the partitions or of the k k equivalence classes of compositions. It extends to a partial order on Λ: for λ,µ ∈ Λ, λ 6 µ ⇔ λ ∼ µ or λ+ < µ+ where λ+,µ+ ∈ Λ+ are the partitions corresponding to λ,µ. (This partial order differs slightly from that used in [2] .) The smallest partition, ν¯ = 1(r), has ν¯ = 1 for all i. The largest partition, i λ=(r), has λ =r,λ =0,i>1. 1 i 7 Write B for either of the R-algebras LGS or RGS . As mentioned above, R R {1 =f(λ,1,λ):λ∈Λ}isasetoforthogonalidempotentsinBwith ⊕ 1 =1 λ λ λ∈Λ (the identity in B). For each partition λ ∈ Λ+ define e = ⊕ 1 and λ µ µ∈Λ,µ<λ e¯ = e ⊕ ⊕ 1 . (For the smallest partition ν¯ = 1(r) define e = 0.) λ λ µ ν¯ µ∈Λ,µ∼λ (cid:18) (cid:19) Note the following results: each e and e¯ is an idempotent; for the largest λ λ partition, (r), we have e¯ = 1 (the identity in B); λ ,λ ∈ Λ+ ⇒ e¯ e¯ = (r) 1 2 λ1 λ2 e¯ and e e =e ; e¯ e =e e¯ =e . min(λ1,λ2) λ1 λ2 min(λ1,λ2) λ λ λ λ λ Obtain a filtration of the algebra B by defining Bλ =e¯ Be¯ ,λ∈Λ+. Then λ λ B(r) = B and λ < λ ⇒ Bλ1 ⊆ Bλ2. Each Bλ is an R-algebra with unit e¯ . 1 2 λ Next define Kλ =(Be B)∩Bλ =Bλe Bλ. Then Kλ is the two-sided ideal in λ λ Bλ generated by e . Finally, define Qλ = Bλ/Kλ. Then Qλ is an R-algebra λ with unit e¯ mod Kλ . λ By standard idempotent theory, if I is an irreducible (left) B-module, then either e¯ I = 0 or e¯ I is an irreducible Bλ(=e¯ Be¯ )-module. Define an irre- λ λ λ λ ducible B-module I to be at level λ if e¯ I 6=0 and e I =0. Then isomorphism λ λ classesofirreducibleB-modulesatlevels6λcorrespondtothosewithe¯ I 6=0, λ which in turn correspond to isomorphism classes of irreducible Bλ-modules. AnirreducibleQλ-modulecorrespondstoanirreducibleBλ-moduleonwhich Kλ acts trivially, and thus to an irreducible B-module I such that e¯ I 6= λ 0 but Kλe¯ I =0. But then Kλe¯ I =0⇒e e¯ I =0⇒e I =0⇒I is at level λ λ λ λ λ λ. Since any irreducible B-module lies at exactly one level, it corresponds to an irreducible Qλ-module for exactly one λ ∈ Λ+. Thus a classification of the irreducible Qλ-modules for all λ∈Λ+ will give a classificationof all irreducible B-modules. We will make one further reduction. Let B¯λ =(1 +e )B(1 +e )⊆Bλ. λ λ λ λ This is an R-algebra with unit 1 +e . Let K¯λ = Kλ ∩B¯λ = B¯λe B¯λ, the λ λ λ two-sided ideal in B¯λ generated by e , and define Cλ = B¯λ/K¯λ. Cλ is an λ R-algebrawith unit 1 modK¯λ and can be identified with a subalgebra of Qλ: λ Cλ = 1 modKλ Qλ 1 modKλ . We will see that irreducible Qλ-modules λ λ correspond to irreducible Cλ-modules. No(cid:0)tice that if µ(cid:1),λ∈(cid:0)Λ and µ ∼(cid:1)λ, then there exists α∈S ⊆S such that r r G =αG α−1. µ λ Lemma 5.1. If µ,λ ∈ Λ, µ ∼ λ, and G = αG α−1, then for X = L or R, µ λ 1 =f(µ,α,λ)∗ 1 ∗ f λ,α−1,µ . µ X λ X Proof. For any ρ∈G we h(cid:0)ave αρα−(cid:1)1 ∈G =G 1G , so λ µ µ µ o(G ), if D =D(µ,1,µ), N D(µ,α,λ),D λ,α−1,µ ,D = λ (0, otherwise. (cid:0) (cid:0) (cid:1) (cid:1) For any σ ∈G , let π =α−1σα∈G . Then σα=απ. Then G (µ,α,λ)=G µ λ R µ and n (µ,α,λ) = o(G ). Reversing the roles of λ,µ gives n λ,α−1,µ = R µ R o(G ). Also n (µ,1,µ) = o(G ). Substituting into the multiplication rule λ R µ (cid:0) (cid:1) 8 then gives f(µ,α,λ)∗ 1 ∗ f λ,α−1,µ =f(µ,α,λ)∗ f λ,α−1,µ R λ R R (cid:0) (cid:1) (cid:0) (cid:1) o(G )o(G ) µ λ = f(µ,1,µ)=1 . µ o(G )o(G ) µ λ By a similar argument, n (µ,α,λ) = o(G ), n λ,α−1,µ = o(G ), and L λ L µ n (µ,1,µ)=o(G ). The multiplication rule again gives L µ (cid:0) (cid:1) f(µ,α,λ)∗ 1 ∗ f λ,α−1,µ =f(µ,α,λ)∗ f λ,α−1,µ L λ L L (cid:0) (cid:1) (cid:0) (cid:1) o(G )o(G ) µ λ = f(µ,1,µ)=1 . µ o(G )o(G ) µ λ Corollary 5.1. 1 modKλ is a “full idempotent” in Qλ, that is, the two sided λ ideal generated by 1 modKλ in Qλ is all of Qλ. λ Proof. Modulo Kλ , e¯ = 1 . By the lemma, the two-sided ideal gen- λ µ µ∈Λ,µ∼λ erated by 1 contains each 1P, µ ∼ λ . So the two sided ideal generated by λ µ 1 modKλ contains the identity e¯ modKλ in Qλ and hence all of Qλ. λ λ Proposition 5.1. Isomorphism classes of irreducible Cλ-modules correspond one to one with isomorphism classes of irreducible Qλ-modules, and hence to isomorphism classes of irreducible B-modules at level λ. Proof. By generalidempotent theory, irreducible Cλ-modules correspondto irreducible Qλ-modules I such that 1 modKλ I 6= 0. But by the corollary λ above, we have a full idempotent, and for a full idempotent 1 modKλ I = (cid:0) (cid:1) λ 0 ⇒ I = 0. So every irreducible Qλ-module corresponds to a Cλ-module as (cid:0) (cid:1) claimed. Remark: Suppose there are d compositions µ in Λ(r) with µ ∼ λ ∈ Λ+(r). It can be shown that Qλ is isomorphic as an R-algebra to the algebra of d by d matrices with entries in Cλ. This leads to an alternative verification of Proposition 5.1. Inthe remainderofthis paper wewill classifythe irreducible B-modules (in many cases) by classifying the irreducible Cλ-modules for all partitions λ. It will be useful to rewrite Cλ slightly: since B¯λ =(1 +e )B(1 +e )=1 B1 +e B1 +1 Be +e Be λ λ λ λ λ λ λ λ λ λ λ λ and e B1 +1 Be +e Be ⊆Be B∩B¯λ =K¯λ λ λ λ λ λ λ λ we have Cλ =B¯λ/K¯λ =1 B1 /(1 B1 ∩Be B) λ λ λ λ λ (where e = 1 ). λ µ µ∈Λ,µ<λ P 9 6 Characteristic zero In this section we will assume R is a field k of characteristic zero. We continue to write B for either of the k-algebras LGS or RGS and let Bλ,Cλ be as in k k section 5. Let ν =1(r) be the “smallest” partition. Theorem 6.1. For a field k of characteristic zero, Cν ∼= k[S ] (the monoid r algebra for S over k) while Cλ =0, λ6=ν. The irreducible left B-modules are r all at level ν and correspond to irreducible left k[S ]-modules. (The irreducible r leftk[S ]-modulesinturncorrespondtoirreducibleleftk[S ]-modulesforvarious r i i6r.) Proof. Let X be either L or R. For the smallest partition ν we have e = ν 0, Cν =1 B1 . ThenCν isak-vectorspacewithbasis{f(ν,D,ν):D ∈ M }. ν ν ν ν Since G = {1}, the ν −ν double cosets D = G αG = α consist of single ν α ν ν elements of S . Then M = S and f(ν,D ,ν) 7→ α gives a vector space r ν ν r α isomorphism φ:Cν →k[S ]. By special case 1 of the multiplication law, r n (D(ν,αβ,ν)) X f(ν,α,ν)∗ f(ν,β,ν)= f(ν,αβ,ν)=f(ν,αβ,ν) X n (D(ν,α,ν)) X (where we used the fact that n (D(ν,γ,ν)) = 1 for any γ ∈ S ). So φ is an X r isomorphism of k-algebras. So irreducible left B-modules at level ν correspond to irreducible left Cν ∼=k[S ]-modules as stated. r Now suppose λ ∈ Λ+, λ 6= ν. To show Cλ = 0 it suffices to prove that 1 B1 ⊆Be B,whichwillbetrueif1 ∈Be B. Wewillshowthat1 ∈B1 B. λ λ λ λ λ λ ν Then since 1 = e¯ = e¯ e , we will have 1 ∈ B1 B = Be¯ e B ⊆ Be B as ν ν ν λ λ ν ν λ λ desired. By special case 4 of the multiplication rule, f(λ,1,ν)∗ 1 ∗ f(ν,1,λ)=f(λ,1,ν)∗ f(ν,1,λ) X ν X X o(G ) λ = f(λ,1,λ)=o(G )1 . λ λ o(G ) ν Since k has characteristic zero, o(S )6=0 in the field k and we have λ 1 1 = f(λ,1,ν)∗ 1 ∗ f(ν,1,λ)∈B1 B λ o(S ) X ν X ν λ as claimed, completing the proof that Cλ =0. 7 Characteristic p We now consider the case where R is a field k with positive characteristic p. Definition 7.1. A partition λ∈Λ+(r) is a p-partition if for each i, 16i6n, either λi =0 or λi =pki for some integer power ki >0 of p. Theorem 7.1. If λ is not a p-partition, then Cλ =0. 10

Generalized Schur Algebras

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