The centers of spin symmetric group algebras and Catalan numbers
aa r X i v : . [ m a t h . C O ] F e b THE CENTERS OF SPIN SYMMETRIC GROUP ALGEBRASAND CATALAN NUMBERS
JILL TYSSE AND WEIQIANG WANG
Abstract.
Generalizing the work of Farahat-Higman on symmetric groups, wedescribe the structures of the even centers Z n of integral spin symmetric groupsuperalgebras, which lead to universal algebras termed as the spin FH-algebras.A connection between the odd Jucys-Murphy elements and the Catalan numbersis developed and then used to determine the algebra generators of the spin FH-algebras and of the even centers Z n . Introduction integral group algebras for the sym-metric groups S n with respect to the basis of conjugacy class sums. This led to twouniversal algebras, K and G , which were then shown to be polynomial algebraswith a distinguished set of ring generators. As an immediate consequence, the cen-ter of the integral group algebra for S n is shown in [2] to have the first n elementarysymmetric polynomials of the Jucys-Murphy elements as its ring generators (whichis a modern reinterpretation since the papers [6, 9] appeared after [2]), and this hasimplications on modular representations of the symmetric groups. A symmetricfunction interpretation of the class sum basis for G was subsequently obtained byMacdonald [8, pp.131-4]. Some results of Farahat-Higman were generalized to thewreath products by the second author and they admit deep connections with thecohomology rings of Hilbert schemes of points on the affine plane or on the minimalresolutions (see Wang [14, 15]).I. Schur in a 1911 paper [12] initiated the spin representation theory of thesymmetric groups by first showing that S n admits double covers e S n , nontrivial for n ≥
4: 1 −→ { , z } −→ e S n −→ S n −→ z is a central element of e S n of order 2. We refer to J´ozefiak [4] (also [5]) for anexcellent modern exposition of Schur’s paper via a systematic use of superalgebras.Given a commutative ring R with no 2-torsion, we form the spin symmetric groupalgebra RS − n = R e S n / h z + 1 i , which has a natural superalgebra structure. Mathematics Subject Classification.
Primary 20C05, Secondary 05E05.
Key words and phrases. spin symmetric groups, Jucys-Murphy elements, Catalan numbers. Z n of the superalgebra RS − n has a basisgiven by the even split class sums of e S n . We first show that the structure constantsin Z n with respect to the class sum basis are polynomials in n and describe asufficient condition for the independence of n of these structure constants, just asin [2]. This leads to two universal algebras, K and F , which we call the filtered andgraded spin FH-algebras respectively. The proofs here are similar to the ones in[2], but we need to carefully keep track of the (sometimes subtle) signs appearingin the multiplications of cycles in RS − n . Our treatment systematically uses thenotion of modified cycle type (cf. [8, p.131]).The odd Jucys-Murphy elements M i for RS − n introduced by Sergeev [13] (also cf.related constructions by Nazarov [11]) will play the role of the usual Jucys-Murphyelements for S n . The odd Jucys-Murphy elements anti-commute with each other,and a conceptual framework has been provided by the notion of degenerate spinaffine Hecke algebras (see Wang [16]). We show that the top degree term of anyelementary symmetric function in the squares M i is a linear combination of theeven split class sums with coefficients given explicitly in terms of the celebratedCatalan numbers, see Theorem 4.5. This remarkable combinatorial connectionis key for establishing the algebraic structure of the spin FH-algebra K . Thenthe algebraic structure of K boils down to some novel combinatorial identity ofCatalan numbers which is verified by using the Lagrange inversion formula. Asa corollary, we obtain a new proof of a theorem in Brundan-Kleshchev [1] for thering generators of RS − n , which has been used in modular spin representations ofthe symmetric groups.Built on the results of Macdonald [8, pp.131-4], we develop a connection betweenthe algebra F and the ring of symmetric functions. This is achieved by establishingan injective algebra homomorphism from F to G .1.3. It is well known that the 2-regular conjugacy classes of the symmetric group S n are parameterized by the odd partitions of n , and so at least formally thereare many similarities between 2-modular representations and complex spin repre-sentations of the symmetric groups. John Murray studied the connections amongFarahat-Higman, Jucys-Murphy, and center of the group algebra F S n , where F isa field of characteristic 2. In particular, a modulo 2 identity of Murray [10, Propo-sition 7.1] which involves the Catalan numbers bears an amazing resemblance toour Theorem 4.5 over integers. It would be very interesting to understand a con-ceptual connection behind these remarkable coincidences between 2-modular andcomplex spin setups for the symmetric groups.1.4. The paper is organized as follows. In Section 2, we review the cycle notationfor elements in the double cover e S n following [4] and the basis of even split classsums for the even center Z n . In Section 3, we establish the basic properties of ENTERS OF SPIN SYMMETRIC GROUP ALGEBRAS 3 the structure constants of Z n , which give rise to the filtered and graded spin FH-algebras. In Section 4, we develop the combinatorial connection between the oddJucys-Murphy elements and Catalan numbers, and then use it in Section 5 toestablish the main structure result for the filtered spin FH-algebra K . Finally inSection 6, we develop a connection between the algebra F and symmetric functions.2. The preliminaries
The double covers of the symmetric groups.
The symmetric group S n is generated by s i , 1 ≤ i ≤ n −
1, subject to the relations s i = 1 , s i s i +1 s i = s i +1 s i s i +1 , s j s i = s i s j for | i − j | > . The generators s i may be identified with the transpositions ( i, i + 1), 1 ≤ i ≤ n − e S n , of S n is defined as the group generated by t , t , . . . , t n − and z , subject to the relations z central , t i = z, z = 1 , t i t i +1 t i = t i +1 t i t i +1 , t j t i = zt i t j for | i − j | > . This gives rise to a short exact sequence of groups1 −→ { , z } −→ e S n θ n −→ S n −→ . where θ n ( t i ) = s i .There is a cycle presentation of e S n , as there is for S n . Following [4], define x i = t i t i +1 · · · t n − t n t n − · · · t i +1 t i ∈ e S n +1 for i = 1 , . . . , n −
1. Then, for a subset { i , . . . , i m } of { , . . . , n } , we define a cycleof length m [ i , i , . . . , i m ] = (cid:26) z, for m = 1 ,x i x i m x i m − · · · x i x i , for m > . It follows that θ n ([ i , . . . , i m ]) = ( i , . . . , i m ) . The even split conjugacy classes of e S n . We sometimes write a parti-tion λ = ( λ , λ , . . . ), a non-increasing sequence of positive integers, or write λ = (1 m , m , . . . ) = ( i m i ) i ≥ , where m i is the number of parts of λ equal to i .We denote the length of λ by ℓ ( λ ) and let | λ | = λ + λ + · · · . Let P ( n ) (respec-tively, E P ( n ), OP ( n )) denote the set of all partitions of n (respectively, having onlyeven, odd parts). Set P = ∪ n ≥ P ( n ), E P = ∪ n ≥ E P ( n ), and OP = ∪ n ≥ OP ( n ).Given w ∈ S n with cycle-type ρ = ( ρ , . . . , ρ t , , . . . , modifiedcycle-type of w to be ˜ ρ = ( ρ − , . . . , ρ t −
1) (see [8, pp.131]). Given a partition λ , let C λ ( n ) denote the conjugacy class of S n containing all elements of modifiedtype λ if | λ | + ℓ ( λ ) ≤ n , and denote C λ ( n ) = ∅ otherwise.What follows is, essentially, a “modified type” version of what appears in [4,Section 3B]. Also we have adopted the more standard modern convention of groupmultiplication from right to left (different from the convention in [12, 4]), e.g.(1 , ,
3) = (1 , ,
3) (instead of (1 , , θ − n ( C λ ( n )) either splits intotwo conjugacy classes of e S n , or it is a single conjugacy class of e S n . In the first case, JILL TYSSE AND WEIQIANG WANG we call C λ ( n ) a split conjugacy class of S n , and call the two conjugacy classes in θ − n ( C λ ( n )) split conjugacy classes of e S n . In fact, we have the following (cf. [4,Theorem 3.6]). Lemma 2.1.
Let C λ ( n ) be a nonempty conjugacy class of S n . Then θ − n ( C λ ( n )) splits into two conjugacy classes of e S n if and only if either (1) λ ∈ E P (whence the conjugacy class or its elements are called even split ),or (2) | λ | is odd, all parts of λ are distinct, and | λ | + ℓ ( λ ) = n or n − . For λ ∈ E P , denote by D λ ( n ) the (even split) conjugacy class in e S n whichcontains the following distinguished element of modified type λt λ = [1 , , . . . , λ +1][ λ +2 , . . . , λ + λ +2] · · · [ λ + . . . + λ ℓ − + ℓ, . . . , λ + . . . + λ ℓ + ℓ ](2.1)and hence θ − n ( C λ ( n )) = D λ ( n ) F zD λ ( n ) if | λ | + ℓ ( λ ) ≤ n ; set D λ ( n ) = ∅ otherwise.Suppose s ′ ∈ θ − n ( s ). Then, the other member of θ − n ( s ) is zs ′ . We denotethe common value of s ′ xs ′− = ( zs ′ ) x ( zs ′ ) − by sxs − , for x ∈ e S n . Let x =[ i , . . . , i m ][ j , . . . , j k ] · · · ∈ e S n be a product of disjoint cycles such that θ n ( x ) is ofmodified type λ . Let s ∈ S n be of modified type µ . Then s ([ i , . . . , i m ][ j , . . . , j k ] · · · ) s − = z | λ || µ | [ s ( i ) , . . . , s ( i m )][ s ( j ) , . . . , s ( j k )] · · · , according to [4, Proposition 3.5]. In particular, we have the following. Lemma 2.2.
For λ ∈ E P of length ℓ , the even split conjugacy class D λ ( n ) consistsof all products of disjoint cycles in e S n of the form [ i , . . . , i λ +1 ][ j , . . . , j λ +1 ] · · · [ k , . . . , k λ ℓ +1 ] . Remark . The D λ ( n ) in the present paper is different from that in [4]. Thecorresponding conjugacy class in [4] - call it ˜ D λ ( n ) to distinguish from our D λ ( n )- is given by ˜ D λ ( n ) = z n − ℓ ( λ ) D λ ( n ) . The present definition of D λ ( n ), just as thedefinition of modified cycle type, is consistent with the natural embedding of e S n in e S n +1 in the sense that D λ ( n + 1) ∩ e S n = D λ ( n ) for all n .The embedding of e S n in e S n +1 gives the union e S ∞ = ∪ n ≥ e S n a natural groupstructure. By Remark 2.3, the union D λ := ∪ n ≥ D λ ( n ) is an even split conjugacyclass in e S ∞ for each λ ∈ E P . We define the homomorphism θ : e S ∞ −→ S ∞ by θ ( x ) = θ n ( x ) for x in e S n . The spin group superalgebra.
Let R be a commutative ring which containsno 2-torsion. The group algebra R e S n has a super (i.e. Z -graded) algebra structuregiven by declaring the elements t i for all 1 ≤ i ≤ n − spin group algebra , RS − n is defined to be the quotient of R e S n by the idealgenerated by z + 1: RS − n = R e S n / h z + 1 i . ENTERS OF SPIN SYMMETRIC GROUP ALGEBRAS 5
We may view the spin group algebra as the algebra generated by the elements t i , i = 1 , . . . , n −
1, subject to the relations t i = − , t i t i +1 t i = t i +1 t i t i +1 , t j t i = − t i t j ( | i − j | > . Remark . One defining relation of RS − n above is t i = − t i = 1 was used in the recent book of Kleshchev [7] and in [16].One can use a scalar √− RS − n adopted in [4] and here differs from the one used in [7, 13] by some scalarsdependent on the length of a cycle.We adapt the notion of cycles for e S n to the spin group algebra setting. As before,and by an abuse of notation, we take x i = t i t i +1 · · · t n − t n − t n − · · · t i +1 t i for i = 1 , · · · , n −
1. Then for a subset { i , . . . , i m } of { , . . . , n } , a cycle in thespin group algebra is defined to be[ i , i , . . . , i m ] = (cid:26) − , for m = 1 ,x i x i m x i m − · · · x i x i , for m > . We have the following properties of the cycles (cf. [4, Theorem 3.1]):[ i , . . . , i m ] = ( − m − [ i , . . . , i m , i ] , [ i, i + 1 , . . . , i + j −
1] = ( − j − t i t i +1 . . . t i + j − , j > , [ a, i , . . . , i m ][ a, j , . . . , j n ] = − [ a, j , . . . , j n , i , . . . , i m ] , if { i , . . . , i m } ∩ { j , . . . , j n } = ∅ . (2.2)The algebra RS − n inherits a superalgebra structure from R e S n by letting theelements t i , 1 ≤ i ≤ n − Z n = Z ( RS − n ) the even centerof RS − n , i.e. the set of even central elements in the superalgebra RS − n , for finite n as well as n = ∞ . For λ ∈ E P , we let d λ ( n ) denote the image in RS − n of the classsum of D λ ( n ) in R e S n if | λ | + ℓ ( λ ) ≤ n , and 0 otherwise. The following is clear. Lemma 2.5.
The set { d λ ( n ) | λ ∈ E P , | λ | + ℓ ( λ ) ≤ n } forms a basis for the evencenter Z n . The structure constants of the centers
Some preparatory lemmas.
Let N be the set of positive integers. For anysubset Y of elements in e S ∞ , define a subset of NN ( Y ) = { j ∈ N | θ ( σ )( j ) = j for some σ in Y } . It is clear that N ( Y ) = ∪ σ ∈ Y N ( σ ). We denote the cardinality of N ( Y ) by | N ( Y ) | . Lemma 3.1. [2, Lemma 3.5]
Let x, y ∈ e S ∞ . Suppose that x is of modified type λ , y is of modified type µ and that xy is of modified type ν . Then | ν | ≤ | λ | + | µ | , withequality if and only if N ( xy ) = N ( x, y ) . JILL TYSSE AND WEIQIANG WANG
Two r -tuples of elements in e S ∞ , ( x , . . . , x r ) and ( y , . . . , y r ), are said to beconjugate in e S ∞ if y i = wx i w − , 1 ≤ i ≤ r for some w in e S ∞ . For any conjugacyclass, C , of such r -tuples, we denote by | N ( C ) | the cardinality of N ( x , . . . , x n ) forany element ( x , . . . , x n ) in C .The next lemma is a spin variant of [2, Lemma 2.1]. Lemma 3.2.
Let C be a conjugacy class of r -tuples of even split elements in e S ∞ .Then, the intersection of C with r z }| {e S n × · · · × e S n is empty if n < | N ( C ) | and is aconjugacy class of r -tuples in e S n if n ≥ | N ( C ) | . The number of r -tuples in theintersection is n ( n − · · · ( n − | N ( C ) | + 1) /k ( C ) where k ( C ) is a constant. Behavior of the structure constants.
Consider the multiplication in theeven center Z n : d λ ( n ) d µ ( n ) = X ν ∈EP a νλµ ( n ) d ν ( n )where the structure constants a νλµ ( n ) are integers and they are undetermined when n < | ν | + ℓ ( ν ). Example 3.3.
Denote by c λ ( n ) the class sum of C λ ( n ) for S n . Then, c (4) (8) c (2) (8) = 25 c (4) (8) + 35 c (2) (8) + 32 c (3 , (8) + 32 c (1 , (8) + 18 c (2 , (8)+7 c (6) (8) + 2 c (4 , (8) ∈ Z ( RS ) ,d (4) (8) d (2) (8) = 13 d (4) (8) − d (2) (8) − d (2 , (8) − d (6) (8) + 2 d (4 , (8) ∈ Z ( RS − ) , [ c (2) (6)] = 2 c (2 , (6) + 5 c (4) (6) + 10 c (2) (6) + 8 c (1 , (6) + 40 c ∅ (6) ∈ Z ( RS ) , [ d (2) (6)] = 2 d (2 , (6) − d (4) (6) + 8 d (2) (6) + 40 d ∅ (6) ∈ Z ( RS − ) . The following is a spin version of [2, Theorem 2.2 and Lemma 3.9], and wepresent its detailed proof to indicate the sign difference from loc. cit.
Theorem 3.4.
Let λ, µ, ν ∈ E P . (1) There is a unique polynomial f νλµ ( x ) such that a νλµ ( n ) = f νλµ ( n ) for all n ≥| ν | + ℓ ( ν ) . The degree of f νλµ ( x ) is no greater than the maximum value of | N ( C ) | − | ν | − ℓ ( ν ) for any class C of pairs ( a, b ) such that a ∈ D λ , b ∈ D µ ,and either ab ∈ D ν or ab ∈ zD ν . (2) The polynomial f νλµ ( x ) = 0 , unless | ν | ≤ | λ | + | µ | . (3) If | ν | = | λ | + | µ | , then the polynomial f νλµ ( x ) is constant, i.e., the structureconstants a νλµ ( n ) are independent of n .Proof. Let ˜ d λ ( n ) be the class sum of D λ ( n ) in R e S n . We write˜ d λ ( n ) ˜ d µ ( n ) = X ν u νλµ ( n ) ˜ d ν ( n ) + X ν v νλµ ( n ) z ˜ d ν ( n ) . Therefore, upon passing to RS − n , we obtain that a νλµ ( n ) = u νλµ ( n ) − v νλµ ( n ) . ENTERS OF SPIN SYMMETRIC GROUP ALGEBRAS 7
For each triple ( λ, µ, ν ) of partitions in
E P , consider the sets X νλµ = { ( a, b ) ∈ e S ∞ × e S ∞ | a ∈ D λ , b ∈ D µ and ab ∈ D ν } ,Y νλµ = { ( a, b ) ∈ e S ∞ × e S ∞ | a ∈ D λ , b ∈ D µ and ab ∈ zD ν } . Any conjugate pair of ( a, b ) ∈ X νλµ (resp. Y νλµ ) also lies in X νλµ (resp. Y νλµ ). There-fore, X νλµ and Y νλµ can be written as disjoint unions of conjugacy classes, say X νλµ = C ⊔ C ⊔ · · · ⊔ C r ,Y νλµ = C r +1 ⊔ C r +2 ⊔ · · · ⊔ C r + s . By Lemma 3.2, the number of pairs ( a, b ) of X νλµ with a and b in e S n is r X i =1 ( n ( n − · · · ( n − | N ( C i ) | + 1)) /k ( C i ) . (3.1)In other words, the above number is equal to the total number of elements from D ν ( n ) that appear upon multiplication of the class sums ˜ d λ ( n ) and ˜ d µ ( n ). To find u νλµ ( n ), we must divide (3.1) by the order of D ν ( n ), which is equal to( n ( n − · · · ( n − | ν | − ℓ ( ν ) + 1)) /k ( ν ) . Note that | ν | + ℓ ( ν ) = | N ( ab ) | ≤ | N ( a, b ) | = | N ( C i ) | for ( a, b ) ∈ C i . We have u νλµ ( n ) = k ( ν ) r X i =1 (cid:0) ( n − | ν | − ℓ ( ν ))( n − | ν | − ℓ ( ν ) − · · · ( n − | N ( C i ) | + 1) (cid:1) /k ( C i ) . One has a similar formula for v νλµ ( n ). So the required polynomial f νλµ ( x ) is givenby r X i =1 k ( ν ) k ( C i ) (cid:0) ( x − | ν | − ℓ ( ν ))( x − | ν | − ℓ ( ν ) − · · · ( x − | N ( C i ) | + 1) (cid:1) − r + s X i = r +1 k ( ν ) k ( C i ) (cid:0) ( x − | ν | − ℓ ( ν ))( x − | ν | − ℓ ( ν ) − · · · ( x − | N ( C i ) | + 1) (cid:1) whose degree is no greater than max ≤ i ≤ r + s {| N ( C i ) | − | ν | − ℓ ( ν ) } . This proves (1).Part (2) holds, since a νλµ ( n ) = 0 for every n unless | ν | ≤ | λ | + | µ | by Lemma 3.1.If | ν | = | λ | + | µ | , it follows by (1) and Lemma 3.1 that the polynomial f νλµ is ofdegree 0, whence a constant. This proves (3). (cid:3) The spin FH-algebras K and F . Let B be the ring consisting of all poly-nomials that take integer values at all integers. By definition, f νλµ ( x ) belongs to B for all λ, µ, ν in E P . We define a B -algebra K with B -basis { d λ | λ ∈ E P} andmultiplication given by d λ d µ = X ν f νλµ ( x ) d ν , where the sum is over all ν in E P such that | ν | ≤ | λ | + | µ | . We will refer to K as the (filtered) spin FH-algebra. The following is an analogue of [2, Theorem 2.4]and it can be proved in the same elementary way.
JILL TYSSE AND WEIQIANG WANG
Proposition 3.5.
The spin FH-algebra K is associative and commutative. Thereexists a surjective homomorphism of algebras φ n : K −→ Z ( Z S − n ) , X f λ ( x ) d λ X f λ ( n ) d λ ( n ) . Let K ( m ) with m being even be the subspace of K that is the B -span of all d λ ’s with | λ | = m (in this case, m will be called the degree of d λ ). Set K ( m ) = ⊕ ≤ i ≤ m,i even K ( i ) . Then, it follows by Theorem 3.4 that {K ( m ) } defines a filteredalgebra structure on K . Given x ∈ K , there is a unique even integer m such that x ∈ K ( m ) and x
6∈ K ( m − , and we denote by x ∗ the top degree part of x such that x − x ∗ ∈ K ( m − . Hence, we have( d λ d µ ) ∗ = X | ν | = | λ | + | µ | f νλµ d ν ∈ K . Define a graded Z -algebra, F ≃ F (0) ⊕ F (2) ⊕ F (4) ⊕ · · · , where F ( m ) ( m even)is defined to be the Z -span of all the symbols d λ with | λ | = m , with multiplication d λ ∗ d µ = X | ν | = | λ | + | µ | f νλµ d ν . Recall that the coefficients f νλµ such that | ν | = | λ | + | µ | (which are equal to a νλµ ( n )for large n ) are integers. Then the graded algebra associated to K is given bygr K ∼ = B ⊗ Z F . It follows from Proposition 3.5 that the algebra F is commutative and associative.We will refer to F as the graded spin FH-algebra. It will be convenient to think of d λ in K or F as the class sum of the conjugacyclass D λ in Z ∞ , but with multiplications modified in two different ways; note thatthe usual multiplication on Z ∞ induced from the group multiplication in e S ∞ hasan obvious divergence problem.Lemma 3.1 also exhibits filtered ring structures on RS − n as well as on Z n , whichallow us to define the associated graded rings gr( RS − n ) and gr Z n . Thus, it willmake sense to talk about the top degree term x ∗ of x ∈ RS − n .3.4. Some distinguished structure constants.
As in [8], we will write λ ∪ µ for the partition that is the union of λ and µ . If µ is a partition contained in λ , thenotation λ − µ will denote the partition obtained by deleting all parts of µ from λ . Let D denote the usual dominance order of partitions. For a one-part partition( s ) with even s , we write d ( s ) = d s . Proposition 3.6.
Let λ = ( i m i ( λ ) ) i ≥ ∈ E P and let s > be even. Then in K , wehave ( d λ d s ) ∗ = X µ ( − ℓ ( µ ) ( m s + | µ | + 1)( s + | µ | + 1) s ! m ( µ )! Q i ≥ m i ( µ )! d λ ∪ ( s + | µ | ) − µ (3.2) where the sum is over all partitions µ = ( i m i ( µ ) ) i ≥ ∈ E P contained in λ with ℓ ( µ ) ≤ ( s + 1) , and m ( µ ) = s + 1 − ℓ ( µ ) . ENTERS OF SPIN SYMMETRIC GROUP ALGEBRAS 9
Proof.
The proof is completely analogous to the proof of [2, Lemma 3.11], andhence will be omitted. Here we only remark that the sign ( − ℓ ( µ ) appearing onthe right-hand side of (3.2) is due to the multiplication of ℓ ( µ ) cycles in d λ withan ( s + 1)-cycle and (2.2). (cid:3) In the same way as Macdonald ([8, pp.132]) reformulated the formula in [2],Proposition 3.6 can be reformulated as follows.
Proposition 3.7.
Let λ = ( i m i ( λ ) ) i ≥ ∈ E P and let s > be even. (1) If | λ | + s = m , then f ( m ) λ ( s ) = ( − ℓ ( λ ) ( m + 1) s !( s + 1 − ℓ ( λ ))! Q i ≥ m i ( λ )! , if ℓ ( λ ) ≤ s + 1 , , otherwise . (2) If | λ | + s = | ν | , and write ν = ( ν , ν , . . . ) , then f νλ ( s ) = X f ( ν i ) µ ( s ) summed over pairs ( i, µ ) such that µ ∪ ν = λ ∪ ( ν i ) , where µ ∈ E P . (3) The coefficient f νλ ( s ) = 0 unless ν D λ ∪ ( s ) , and f λ ∪ ( s ) λ ( s ) > .Remark . Let λ = ( i m i ( λ ) ) i ≥ and µ = ( i m i ( µ ) ) i ≥ be partitions in E P . As avariant of [2, Lemma 3.10], we have the following formula of structure constants: f λ ∪ µλµ = Y i ≥ ( m i ( λ ) + m i ( µ ))! m i ( λ )! m i ( µ )! . Theorem 3.9.
The algebra Q ⊗ Z F is a polynomial algebra generated by d m with m = 2 , , , . . . . Proof.
Let λ = ( λ , λ , . . . ) ∈ E P . By Proposition 3.7, we have inductively d λ ∗ d λ ∗ · · · = X µ ∈EP ,µ D λ h λµ d µ (3.3)where h λµ ∈ Z with h λλ > h λµ ) is a triangular integralmatrix with nonzero diagonal entries, whose inverse matrix has rational entries. (cid:3) The Jucys-Murphy elements and Catalan numbers
The odd Jucys-Murphy elements.
Recall the Jucys-Murphy elements [6,9] in the symmetric group algebra RS n are defined to be ξ k = P k − i =1 ( i, k ) for1 ≤ k ≤ n. The odd Jucys-Murphy elements M k in the spin group algebra RS − n were introduced by Sergeev [13] up to a common factor √− M k = k − X i =1 [ i, k ] , ≤ k ≤ n and they are closely related to the constructions in [11, 16]. It follows by (2.2) that M k = − ( k − − k − X i,j =1; i = j [ i, j, k ] . (4.1)We caution that our M k differs from the square of the odd Jucys-Murphy elementsused in [1, 7, 13, 16] by a sign.According to [11, 13], for 1 ≤ r ≤ n , the r th elementary symmetric function inthe M k (1 ≤ k ≤ n ) e r ; n = X ≤ i
The first few e ∗ r are computed as follows. e ∗ = − d e ∗ = d (2 , − d e ∗ = − d (2 , , + 2 d (4 , − d . The relations among A λ .Lemma 4.2. Let r ≥ and λ = ( λ , λ , . . . , λ ℓ ) ∈ E P (2 r ) be of length ℓ. Thecoefficients A λ in (4.2) admit the following factorization property: A λ = A λ A λ · · · A λ ℓ . Proof.
The proof is by induction on ℓ ( λ ) = ℓ , with the case ℓ ( λ ) = 1 being trivial.Let λ ∈ E P (2 r ) be of length ℓ >
1. Recall that the element t λ in (2.1) of modifiedtype λ = ( λ , λ , . . . , λ ℓ ) can be written as t λ = uv , where u = [1 , , . . . , λ + 1] · · · [ λ + · · · + λ ℓ − + ℓ − , . . . , λ + · · · + λ ℓ − + ℓ − λ = ( λ , λ , . . . , λ ℓ − ) and length ( ℓ − v = [ λ + · · · + λ ℓ − + ℓ, . . . , λ + · · · + λ ℓ + ℓ ] . To find the coefficient A λ of d λ in e ∗ r we will count the number of appearances of t λ in e ∗ r . Suppose that t λ appears upon multiplication of a particular term, M i · · · M i r ,of e ∗ r . Since t λ has the top degree 2 r among the cycles in M i · · · M i r , we can andwill regard the products M i s and M i · · · M i r as in gr( RS − n ) in the remainder of this ENTERS OF SPIN SYMMETRIC GROUP ALGEBRAS 11 proof, e.g. M k = − P ≤ i = j ≤ k − [ i, j, k ]. To produce a (2 n + 1)-cycle by multiplyingtogether 3-cycles, one needs n i r = λ + · · · + λ ℓ + ℓ .Furthermore, since i < · · · < i r and because of the increasing arrangement of thecycles of u and v , the cycle v must appear upon multiplication of the last λ ℓ / M i k , and the smallest such i k has to satisfy i k ≥ λ + · · · + λ ℓ − + ℓ + 2. Thefactor u must appear upon multiplication of the first ( λ + · · · + λ ℓ − ) / M i k with the largest such i k being λ + · · · + λ ℓ − + ℓ − t λ in e r comes exactly from thepartial sum P M i · · · M i r of e r where the indices satisfy the conditions3 ≤ i < i < · · · < (cid:0) i r − λ ℓ = λ + · · · + λ ℓ − + ℓ − (cid:1) < i r − λ ℓ +1 < · · · < i r = λ + · · · + λ ℓ + ℓ and i r − λ ℓ / ≥ λ + · · · + λ ℓ − + ℓ + 2. This partial sum can be written as theproduct E E , where E = X M i · · · M i r − λℓ , E = X M i r − λℓ +1 · · · M i r . Thus, the multiplicity of t λ in e ∗ r is the product of the coefficient of u in E , whichis Q ℓ − i =1 A λ i by the induction assumption, and the coefficient of v in E , which is A λ ℓ . This proves the lemma. (cid:3) Lemma 4.3.
The following recursive relation holds: A = − , A r = 2 A r − − r − X s =1 A (2 r − − s, s ) ( r ≥
2) (4.3) where we have denoted A ( b,a ) = A ( a,b ) for a > b .Proof. We have seen that A = − A r is equal tothe coefficient of d (2 r ) (2 r + 1) appearing in e ∗ r , which is the multiplicity of each(2 r + 1)-cycle, say [1 , , · · · , r + 1], that appears in e r ;2 r +1 . Observe that only thefollowing summand X ≤ i < ···
2) satisfies | λ | + ℓ ( λ ) = 2 r − ℓ ( λ ) ≤ r , that is, ℓ ( λ ) ≤
2. Therefore, any such λ must beone of the following partitions λ ( s ) := (2 r − − s, s ) , ≤ s ≤ r − . The coefficient of d λ ( s ) (2 r ) in P i < ···
We have the following recursive relation for the Catalan numbers: C = 1 , C r +1 = 2 C r + r − X s =1 C r − s C s +1 ( r ≥ . (4.6) ENTERS OF SPIN SYMMETRIC GROUP ALGEBRAS 13
Theorem 4.5.
Let λ = ( λ , λ , . . . ) ∈ E P . The coefficients A λ are given by: A r = − C r +1 ,A λ = A λ A λ · · · = ( − ℓ ( λ ) Y i ≥ C λi +1 . Proof.
By Lemma 4.2 and Lemma 4.3, we have A = − , A r = 2 A r − − r − X s =1 A r − − s A s ( r ≥ . This implies that − A r satisfies the same initial condition for r = 1 and the samerecursive relation as for C r +1 by Lemma 4.4, whence − A r = C r +1 . The generalformula for A λ follows from this and Lemma 4.2. (cid:3) The structures of the algebra K and the centers Z n A criterion for the subspace H ( m ) . Given an even positive integer m ,let H ( m ) denote the B -submodule of K ( m ) spanned by the elements ( d λ d µ ) ∗ where λ, µ ∈ E P , | λ | + | µ | = m , | λ | >
0, and | µ | > ν = (2 m , m , m , . . . ) in E P we define the following polyno-mial (which lies in B ): P ν ( x ) = ( − ℓ ( ν ) (cid:18) xm , m , m , . . . (cid:19) = ( − ℓ ( ν ) x ( x − · · · ( x − ( m + m + m + . . . ) + 1) m ! m ! m ! . . . . The proposition below is an analogue of [2, Theorem 4.3], and it can be provedas in loc. cit. . We remark that the sign in the definition of the polynomial P ν ( x ),different from loc. cit. , has its origin in the formula of Proposition 3.6 which isused in the proof in an essential way. Proposition 5.1.
An element P ν ∈EP ( m ) a ν d ν of K ( m ) is contained in H ( m ) if andonly if X ν ∈EP ( m ) a ν P ν ( − m ) = 0 . (5.1)5.2. The algebra generators.
We shall need the Lagrange inversion formula (cf.e.g. [3, Sect. 1.2.4]) which we recall here. Let S [[ x ]] denote the set of all formalpower series in the variable x with coefficients in a commutative ring S . Let S [[ x ]] denote the subset of all formal power series in the variable x that have zero constantterm and let S [[ x ]] denote the subset of all power series that have nonzero constantterm. Let S (( x )) denote the set of all formal Laurent series in x with coefficientsin S . Finally, given a power series or Laurent series f , we define [ x i ] f to be thecoefficient of x i in the series f . Lemma 5.2 (Lagrange inversion formula) . Let φ ( s ) ∈ S [[ s ]] . Then there existsa unique formal power series w ( x ) ∈ S [[ x ]] such that w = xφ ( w ) . Moreover, if f ( s ) ∈ S (( s )) , then [ x n ] f ( w ) = 1 n [ s n − ] { f ′ ( s ) φ n ( s ) } , for n = 0 . Theorem 5.3.
Let m = 2 r be an even positive integer. Then the coefficients A λ from (4.2) satisfy the identity X λ ∈EP ( m ) A λ P λ ( − m ) = 2( − r . Proof.
We calculate by Theorem 4.5 and the binomial theorem that P λ ∈EP ( m ) A λ P λ ( − m )= [ y m ] X λ =(2 i , i , i , ··· ) ( − ℓ ( λ ) A λ (cid:18) − mi , i , i , · · · (cid:19) y i +4 i +6 i + ··· = [ y m ] X N ≥ ( − N X N = i + i + ··· A i A i A i · · · (cid:18) − mi , i , i , · · · (cid:19) y i y i y i · · · = [ y m ] X N ≥ ( − N (cid:18) − mN (cid:19) ( A y + A y + A y + · · · ) N , which, using A r = − C r +1 from Theorem 4.5 and (4.4–4.5), is= [ y m ] X N ≥ ( − N (cid:18) − mN (cid:19)(cid:0) y − c ( y ) (cid:1) N = [ y m ] (cid:0) − (1 + y − c ( y ) (cid:1) − m = [ y m ] (cid:0) − c ( y ) (cid:1) m . Write m = 2 r and x = y . Now the proof of the theorem is completed byLemma 5.4 below. (cid:3) Lemma 5.4.
Let r be a positive integer. Then, [ x r ](1 − c ( x )) r = 2( − r . Proof.
We rewrite (4.5) as c ( x ) = x (1 − c ( x )) − . Applying the Lagrange inversionformula in Lemma 5.2 (by setting w = c and φ = (1 − c ) − therein) gives us[ x r ] c ( x ) k = 1 r [ s r − ] { ks k − (1 − s ) − r } = kr [ s r − k ](1 − s ) − r = kr ( − r − k (cid:18) − rr − k (cid:19) . ENTERS OF SPIN SYMMETRIC GROUP ALGEBRAS 15
Hence, by the binomial theorem and noting that [ x r ] c ( x ) k = 0 for k > r , we have[ x r ](1 − c ( x )) r = r X k =1 (cid:18) rk (cid:19) [ x r ]( − c ( x )) k = r X k =1 ( − r kr (cid:18) rk (cid:19)(cid:18) − rr − k (cid:19) = 2( − r r X k =1 (cid:18) r − k − (cid:19)(cid:18) − rr − k (cid:19) = 2( − r (cid:18) r − r − (cid:19) = 2( − r . In the second last equality, we have used a standard binomial formula: for c ≥ c X s =0 (cid:18) as (cid:19)(cid:18) bc − s (cid:19) = (cid:18) a + bc (cid:19) . (cid:3) Denote by B [ ] the localization at 2 of the ring B . Set K [ ] = B [ ] ⊗ B K and K [ ] ( m ) = B [ ] ⊗ B K ( m ) for m even. Theorem 5.5. (1)
Let m = 2 r ∈ N . Then, the B [ ] -module K [ ] ( m ) is spannedby e ∗ r and H ( m ) . (2) As a B [ ] -algebra, K [ ] is generated by e ∗ r for r ≥ .Proof. (1) Let X = P B λ d λ be an element of K ( m ) , and write P B λ P λ ( − m ) = q for some q ∈ B . Recall e ∗ r = P λ ∈EP (2 r ) A λ d λ . We have P λ A λ P λ ( − m ) = 2( − r ,Theorem 5.3. Then, Y := 2 X − ( − r qe ∗ r ∈ H ( m ) by Proposition 5.1, and hence X = ( − r q e ∗ r + Y lies in the span of e ∗ r and H ( m ) .(2) Let A be the B [ ]-subalgebra of K [ ] generated by e ∗ , e ∗ , . . . . It suffices toshow that K ⊆ A , or K ( m ) ⊆ A for all even m ≥ m . Certainly, K (0) ⊆ A . Let m = 2 r > K ( n ) ⊆ A for all even n < m . Thisimplies that H ( m ) is contained in A by the definition of H ( m ) . Since by definition A also contains e ∗ r , we have by (1) that K ( m ) is contained in A . (cid:3) As a corollary to Theorem 5.5, we have the following theorem by applying thesurjective homomorphism φ n : K [ ] −→ Z ( Z [ ] S − n ) (see Proposition 3.5) and abase ring change. Another proof based on Murphy’s method (cf. [9]) of Theo-rem 5.6 was given earlier by Brundan and Kleshchev [1] (whose working assumptionthat R is a field of characteristic = 2 can be obviously relaxed). Theorem 5.6.
Let R be a commutative ring which contains (e.g. any fieldof characteristic not equal to ). Then the even center of RS − n is the R -algebragenerated by (the top-degree terms of ) the r th elementary symmetric functions in M , . . . , M n with r = 1 , . . . , n . We remark that “the top-degree terms of” in the statement of the above Theoremis easily removable, if one follows the proof of Theorem 5.5 (1) more closely togetherwith the surjective homomorphism φ n .6. Connections with symmetric functions
Recall that the original results for the symmetric groups similar to our Theo-rem 3.4 and Proposition 3.6 were established in Farahat-Higman [2]. This led tothe introduction of a ring G by Macdonald (see [8, pp.131–134]), which is com-pletely analogous to our current F . Recall that G is the Z -span of c λ for λ ∈ P ,where c λ is formally the class sum of the conjugacy class in S ∞ of modified type λ (analogous to our d λ ). The algebra Q ⊗ Z G is known to be freely generated by c r ,r = 1 , , , . . . , where c r = c ( r ) . Let Λ denote the ring of symmetric functions over Z . Macdonald then established a ring isomorphism ϕ : G → Λ, c λ g λ , where g λ is a new basis of symmetric functions explicitly defined in [8, pp.132–134]. Proposition 6.1. (1)
There exists a natural injective algebra homomorphism ι : F −→ G, which sends d λ ( − ℓ ( λ ) c λ for each λ ∈ E P . (2) There exists an isomorphism of algebras ψ : Q ⊗ Z F → Λ e Q := Q [ p , p , . . . ] which sends d λ to ( − ℓ ( λ ) g λ for each λ ∈ E P .Proof. (1) The multiplication in the ring G is written as c λ ∗ c µ = X ν ∈P : | ν | = | λ | + | µ | k νλµ c ν , for λ, µ ∈ P . By [2, Lemma 3.11] (or [8, pp.132, (5), (6)] where the notation a νλµ was used for k νλµ ), we observe that, for λ ∈ E P and s an even integer, the constant k νλ ( s ) = 0unless ν ∈ E P and | ν | = | λ | + s . By further comparing with Proposition 3.7, wehave k νλ ( s ) = ( − ℓ ( λ )+1 − ℓ ( ν ) f νλ ( s ) , which is equivalent to ι ( d λ ∗ d s ) = ( − ℓ ( λ )+1 c λ ∗ c s = ι ( d λ ) ∗ ι ( d s ) . (6.1)This and (3.3) imply that for λ = ( λ , λ , . . . ) ∈ E P , c λ ∗ c λ ∗ · · · = X µ ∈EP ,µ D λ ( − ℓ ( λ )+ ℓ ( µ ) h λµ c µ . Hence, for µ ∈ E P ,( − ℓ ( µ ) c µ = X λ ∈EP ˜ h µλ ( − c λ ) ∗ ( − c λ ) ∗ · · · d µ = X λ ∈EP ˜ h µλ d λ ∗ d λ ∗ · · · , where [˜ h µλ ] denotes the inverse matrix of [ h λµ ]. Now it follows from these identitiesand (6.1) that ι ( d λ ∗ d µ ) = ι ( d λ ) ∗ ι ( d µ ), i.e., ι is an algebra homomorphism.It follows from (1) and Theorem 3.9 that the image ι ( Q ⊗ Z F ) is the polynomialalgebra generated by c ( s ) , s = 2 , , , . . . . Now (2) follows by composing ι and the ENTERS OF SPIN SYMMETRIC GROUP ALGEBRAS 17 ring isomorphism ϕ : G → Λ, c λ g λ and by noting that g ( s ) = − p s for all (even) s ≥ (cid:3) Proposition 6.1 (1) is essentially equivalent to the claim that there is no can-cellation in the contributions to d ν when the (spin) permutations are multipliedbetween the class sums d λ and d µ . It fits with the computations in Example 3.3. Remark . In the same way as e ∗ r was defined right before (4.2), we can define p ∗ r ∈ F as the “ ∗ -stabilization” of the r th power sum of M , M , M , . . . . Then onecan show as in [15, Propositions 5.5, 5.6] that the isomorphism ψ : Q ⊗ Z F → Λ e Q sends p ∗ r to − p r . If we denote by η the algebra isomorphism Λ e Q → Λ Q := Q ⊗ Z Λsending p r to p r for each r , then the composition ηψ sends e ∗ r to ( − r h r , where h r denotes the r th complete homogeneous symmetric function. Problem 1.
Determine explicitly the distinguished basis for Λ Q which correspondsto the basis d λ in Q ⊗ Z F under the isomorphism ηψ . In the original setup of symmetric groups [2, 8], Macdonald’s isomorphism G ∼ = Λactually identifies the r th elementary symmetric function in the Jucys-Murphy el-ements with ( − r h r ∈ Λ, according to [15, Theorem 5.7] (see also [10, Propo-sition 3.2]). So the symmetric functions in the above Problem can be viewed asanother reasonable spin analogue of Macdonald’s symmetric functions g λ . Acknowledgment.
The research of W.W. is partially supported by the NSAand NSF grants. We thank John Murray for bringing his very interesting paper toour attention when we posted an earlier version of this paper in the arXiv.
References [1] J. Brundan and A. Kleshchev,
Representation theory of symmetric groups and their doublecovers , In: Groups, combinatorics & geometry (Durham, 2001) , 31–53, World Sci. Publ.,2003.[2] H. Farahat and G. Higman,
The centres of symmetric group rings , Proc. Roy. Soc. (A) (1959), 212–221.[3] I. Goulden and D. Jackson,
Combinatorial Enumeration , Wiley-Interscience Series in Dis-crete Math., 1983.[4] T. J´ozefiak,
Characters of projective representations of symmetric groups , Expo. Math. (1989), 193–247.[5] T. J´ozefiak, Semisimple superalgebras , In: Algebra–Some Current Trends (Varna, 1986),Lect. Notes in Math. (1988), 96–113, Springer-Berlag, Berlin-New York.[6] A. Jucys,
Symmetric polynomials and the center of the symmetric group rings , Rep. Math.Phys. (1974), 107–112.[7] A. Kleshchev, Linear and projective representations of symmetric groups , CambridgeTracts in Mathematics , Cambridge University Press, 2005.[8] I.G. Macdonald,
Symmetric functions and Hall polynomials , Second Ed., Clarendon Press,Oxford, 1995.[9] G. Murphy,
A new construction of Young’s seminormal representation of the symmetricgroup , J. Algebra (1981), 287–291.[10] J. Murray, Generators for the centre of the group algebra of a symmetric group , J. Algebra (2004), 725–748. [11] M. Nazarov,
Young’s symmetrizers for projective representations of the symmetric group ,Adv. in Math. (1997), 190–257.[12] I. Schur, ¨Uber die Darstellung der symmetrischen und der alternierenden Gruppe durchgebrochene lineare Substitutionen, J. Reine Angew. Math. (1911), 155-250.[13] A. Sergeev,
The Howe duality and the projective representations of symmetric groups ,Represent. Theory (1999), 416–434.[14] W. Wang, The Farahat-Higman ring of wreath products and Hilbert schemes , Adv. inMath. (2004), 417–446.[15] W. Wang,
Universal rings arising in geometry and group theory , In: S.D. Cutkosky,D. Edidin, Z. Qin and Q. Zhang (eds.), Vector bundles and representation theory, Con-temp. Math. (2003), 125–140.[16] W. Wang,
Double affine Hecke algebras for the spin symmetric group , preprint 2006,math.RT/0608074.
Department of Mathematics, University of Virginia, Charlottesville, VA 22904
E-mail address ::