Computing Cocharacters of Sign Trace Identities in Reduced Notation

Computing Cocharacters of Sign Trace Identities in Reduced Notation Luisa Carini Dipartimento di Matematica, Universit`a di Messina, Salita Sperone 31, 98166, Messina, Italy. E-mail address: [email protected]

Abstract In [11], Regev applied the representation theory of a general Lie superalgebra to generalize the theory of trace identities as developed by Procesi and Razmyslov. Regev showed that certain chocaracters arising from sign trace identities were given by X

χλ ⊗ χλ

λ∈H(k;l;n)

where χλ ⊗ χλ denotes the Kronecker product of the irreducible character of the symmetric group associated with the partition λ with itself and H(k; l; n) denotes the set of partitions of n λ = (λ1 ≥ λ2 ≥ ... ≥ λn ) such that λk+1 ≤ l. In case of k = 2, l = 1, we compute some multiplicities which occur in the expansion of the cocharacter in terms of irreducible characters by using the reduced notation [13].

AMS classification: 16R30 Keywords: trace identities, invariant theory, Kronecker product, Schur functions

The theory of trace identities, developed indipendently by Procesi[8] and Razmyslov [9] has proved to be a powerful tool in the study of identities of the algebra Mk (F ) of k × k matrices over a field F of characteristic 0. In [11] Regev has given a ”hook” generalization of the theory of trace identities, which has applications to the study of certain P.I. algebras. Briefly in the 1

usual theory of trace identities, the group algebra C(Sn ) of the symmetric group is identified with multilinear trace polynomials. Then one can use the classical work of Schur and Weyl [14] on the polynomial representations of the general linear algebra gl(k, C) and show that the trace cocharacter of P the Mk (F ) equals λ∈Λk (n) χλ ⊗ χλ , where ⊗ denotes the Kronecker or inner product of the irreducible Sn −character χλ with itself and Λk (n) denotes the set of partitions of n with k or fewer parts. It follows from the basic properties of the Kronecker products that: X

χλ ⊗ χλ =

λ∈Λk (n)

X

mµ (Mk (F ))χµ .

µ∈Λk2 (n)

The multiplicities mµ (Mk (F )) are non negative integers and are explicitly known only for k = 2 and partially for k = 3 (see [1]) and they are not yet well understood. In [11] Regev defines a certain quasi Z2 -grading on Mk+1 (F ) depending on k and l and he denotes the resulting (k, l) quasi-structure by Mk,l (F ). One can then define by analogy the n-th sign trace cocharacter of Mk,l (F ), denoted by χST n (Mk,l (F )), and as one of the major results of [11], (M Regev proves that χST k,l (F )) equals n X

χλ ⊗ χλ

(1)

λ∈H(k;l;n)

where H(k; l; n) denotes the set of partitions λ = (λ1 ≥ λ2 ≥ ... ≥ λn ≥ 0) where λk+1 ≤ l. This character is associated with several objects in PI theory: with sign trace identities, with the PI’s of the identities of the 3x3 matrices with the (2,1) superalgebra structure (see [10]) and it is also related to the cocharacters of the ordinary 3x3 matrices (see [1]). In [12] Remmel gives an explicit formula for (1) in the case where k = l = 1 and he proves the following theorem: Theorem 1 Let X

χλ ⊗ χλ =

λ∈H(1;1;n)

n X

χ(r,1n−r ) ⊗ χ(r,1n−r ) =

X

cµ χµ .

µ

r=1

Then 1. cµ = 0 if µ is not a hook or a double hook shape; (

2. c(r,1n−r ) =

r if n-r is even r − 1 if n-r is odd

3. c(q,p,2b ,1a ) = 2(q − p + 1) if q ≥ p ≥ 2 and q + p + 2b + a = n. 2

The purpose of this paper is to compute, in the case where k = 2, l = 1, some multiplicities which occur in the expansion of (1) in terms of irreducible characters. More precisely, we prove the following Proposition. Let χST n (M2,1 ) =

X

χλ ⊗ χλ =

X

cµ χ µ

(2)

µ

λ∈H(2;1;n)

where H(2, 1, n) = {λ ` n : λ3 ≤ 1}. Then n even: • c(n−1,1) =

(n−2)2 2

• c(n−2,2) =

3n2 −19n+34 2

• c(n−2,12 ) = n2 − 6n + 10; n odd: • c(n−1,1) =

n2 −4n+5 2

• c(n−2,2) =

3n2 −19n+32 2

• c(n−2,12 ) = n2 − 6n + 10. Conjecture. If we expand (2) for up to n = 17, 18 by using the computer package SCHUR, it is noticeable and it may stated as a conjecture, the stabilization of coefficients as the column length increases. Thus the coefficient c(1n ) stabilises at n = 3, c(2,1n−2 ) at n = 6 and generally c(k,1n−k ) stabilises at n = 3k and it is equal to k3 (2k 2 − 3k + 4). One then notices that c(k,2,1n−2−k ) stabilises at n = 3k + 2, likewise c(k,3,1n−3−k ) stabilises for k = 3 at n = 13, k = 4 at n = 16. Steps of 3 seem to be relevant. In our computation we use the reduced notation for irreducible representations of the symmetric group Sn [13], which is a powerful tool for investigating the stability of diverse properties of the symmetric group such as Kronecker products, irreducible characters, inner plethysms . The outline of this paper is as follows. In Section 1 and 2 we shall state some preliminaries on reduced notation and Littlewood’s modification rules. Then in Section 3, we shall apply this method to carry out our main computation. 3

A remark about notation. In this paper, we shall freely mix the traditional notation of Littlewood with that of Macdonald [6], which is more convenient for algebraic manipulations. So, the Schur function corresponding to a partition λ will be indifferently denoted by {λ} or sλ .

1

Reduced notation

The concept of reduced notation for the symmetric group was introduced by Murnaghan in 1938 [7] and later used by Littlewood [4,5] for the calculation of inner plethysm and Kronecker products for the symmetric group Sn . The irreducible representation {λ} of Sn may be labelled by ordered partitions (λ) of integers where λ ` n. In reduced notation the label {λ} = {λ1 , λ2 , ..., λp } for Sn is replaced by < λ >=< λ2 , .., λp >. Given any irreducible representation < µ > in reduced notation it can be converted back into a standard irreducible representation of Sn by prefixing < µ > with the integer (n − |µ|). For example, an irreducible representation < 2, 1 > in reduced notation corresponds to {3, 2, 1} in S6 or {9, 2, 1} in S12 . It is just this feature that leads to an n-independent notation for Sn . If n − |µ| ≥ µ1 , then the resulting irreducible representation {n − |µ|, µ} is assuredly a standard irreducible representation of Sn . However, if n − |µ| < µ1 , then the irreducible representation {n − |µ|, µ} is non standard and must be converted into standard form using the following s-function modification rules (Littlewood 1950): • In any s-function two consecutive parts may be interchanged provided that the preceding part is decreased by unity and the succeeding part increased by unity, the resulting s-function being thereby changed in sign, i.e. {λ1 , .., λi , λi+1 , .., λk } = −{λ1 , .., λi+1 − 1, λi + 1, ..., λk } • In any s-function if any part exceeds by unity the preceding part, the value of s-function is zero, i.e. if λi+1 = λi +1 then {λ1 , .., λi , λi+1 , .., λk } = 0 • The value of any s-function is zero if the last part is negative. Examples: 4

(a) Consider in reduced notation < µ >=< 2, 1 >; in S3 µ =< 2, 1 > becomes {n − |µ|, µ} = {0, 2, 1} which is not standard and must be made standard using the above Littlewood’s modification rules. Therefore by (1) we get {0, 2, 1} = −{1, 1, 1} = −{13 }, Instead in S4 , µ =< 2, 1 > becomes {n − |µ|, 2, 1} = {1, 2, 1} and by (2) we get {1, 2, 1} = 0. (b) < 4, 2 > in S8 becomes {2, 4, 2} = −{3, 3, 2} while in S9 we get {3, 4, 2} which is zero by (2); in S4 , µ =< 4, 2 > becomes {n − |µ|, 4, 2} = {−2, 4, 2} and by applying (a) twice we get: {−2, 4, 2} = −{3, −1, 2} = {3, 1, 0} = {3, 1}.

2

Reduced Kronecker products

A reduced Kronecker product < λ > ◦ < µ > may be evaluated by the recursive relation (see [4]) < λ > ◦ < µ >=

X

< {λ/αβ} · {µ/αγ} · {β ◦ γ} >

α,β,γ

where ”/” indicates an s-function skew, i.e. {λ/µ} = Dsµ sλ , (see [5]), a dot is for Littlewood-Richardson s-function multiplication and ” ◦ ” is the ordinary inner (Kronecker) product. By the notation {λ/αβ} and {µ/αγ} we mean the Schur functions corresponding to all those partitions obtained by removing all allowed β and γ with the same weight, (i.e.same number of cells in their corresponding diagrams),from the skew diagrams λ/α and µ/α for all possible partitions α contained in λ. We show in the following detailed example, how the above formula works: Calculation of < 2, 1 > ◦ < 22 > < 2, 1 > ◦ < 22 >=

X

< 2, 1/αβ · 22 /αγ · β ◦ γ >

βγ

Choises for α are 0, 1, 12 , 2, 21. Thus (3) becomes < 21 > ◦ < 22 >= X

< 21/β · 22 /γ · β ◦ γ >

(4a)

βγ

+

X

< 2 + 12 /β · 21/γ · β ◦ γ >

βγ

5

(4b)

(3)

+

X

< 1/β · 12 /γ · β ◦ γ >

(4c)

< 1/β · 2/γ · β ◦ γ >

(4d)

< 0/β · 1/γ · β ◦ γ >

(4e)

βγ

+

X βγ

+

X βγ

Now evaluate (4a)-(4e) for the allowed values of β, γ, thus (4a) involves the terms β = γ = 0 < 2, 1·22 >=< 4, 3 > + < 4, 2, 1 > + < 32 , 1 > + < 3, 22 > + < 3, 2, 12 > + < 23 , 1 > β = γ = 1 < 2+12 ·21·1 >=< 5, 1 > +3 < 4, 2 > +3 < 4, 12 > +2 < 32 > +6 < 3, 2, 1 > +3 < 3, 13 > +2 < 23 > +3 < 22 , 12 > + < 2, 14 > β = γ = 2 < 1·2·2 >=< 5 > +2 < 4, 1 > +2 < 3, 2 > + < 3, 12 > + < 22 , 1 > β = γ = 12 < 1·12 ·2 >=< 4, 1 > + < 3, 2 > +2 < 3, 12 > + < 22 , 1 > + < 2, 13 > β = 2, γ = 12 < 1·12 ·12 >=< 3, 2 > + < 3, 12 > +2 < 22 , 1 > +2 < 2, 13 > + < 15 > β = 12 , γ = 2 < 1·2·12 >=< 4, 1 > + < 3, 2 > +2 < 3, 12 > + < 22 , 1 > + < 2, 13 > β = γ = 2, 1 < 0·1·2, 1◦2, 1 >=< 4 > +2 < 3, 1 > + < 22 > +2 < 2, 12 > + < 14 > Then for (4b) β = γ = 0 < 2+12 ·21·0 >=< 4, 1 > +2 < 3, 2 > +2 < 3, 12 > +2 < 22 , 1 > + < 2, 13 > β = γ = 1 < 2·1·2+12 ·1 >= 2 < 4 > +6 < 3, 1 > +4 < 22 > +6 < 2, 12 > +2 < 14 > β = γ = 2 < 0 · 1 · 2 >=< 3 > + < 2, 1 > β = γ = 12 < 0 · 1 · 2 >=< 3 > + < 2, 1 > β = 2, γ = 12 < 0 · 1 · 12 >=< 2, 1 > + < 13 > (4b0 )

β = 12 , γ = 2 < 0 · 1 · 12 >=< 2, 1 > + < 13 > Continuing for (4c) β = γ = 0 < 1 · 2 · 0 >=< 3 > + < 2, 1 > β = γ = 1 < 0 · 1 · 1 >=< 2 > + < 12 >

(4c0 )

and for (4d) β = γ = 0 < 1 · 12 · 0 >=< 13 > + < 2, 1 > β = γ = 1 < 0 · 1 · 1 >=< 2 > + < 12 > 6

(4d0 )

(4a0 )

and for (4e) (4e0 )

β = γ = 0 < 0 · 1 · 0 >=< 1 > Adding up the results for (4a’) to (4e’) yields < 2, 1 > ◦ < 22 > = < 5, 1 > +3 < 4, 12 > + < 3, 22 > +8 < 3, 12 > +3 < 22 , 12 > +8 < 2, 12 > +3 < 13 >

3

+ +5 < 4, 1 > < 3, 2, 12 > +8 < 3, 1 > +7 < 22 , 1 > +6 < 2, 1 > +2 < 12 >

+ < 4, 3 > +3 < 4 > +6 < 3, 2, 1 > +3 < 3 > +5 < 22 > +2 < 2 > +

+ < 4, 2, 1 > + < 32 , 1 > +7 < 3, 2 > + < 23 , 1 > + < 2, 14 > + < 15 >

The sign trace cocharacter of M2,1(F )

In [11] Regev proves that the sign trace chocaracter of Mk,l (F ), χST n (Mk,l (F )) equals X X χλ ⊗ χλ = mµ χµ . µ

λ∈H(k,l,n)

where H(k, l, n) denotes the set of partition λ = (λ1 ≥ λ2 ≥ .... ≥ λn ≥ 0), where λk+1 ≤ l. In this section we consider the case of k = 2, l = 1 and compute some multiplicities which occur in the expansion of the n-th sign trace cocharacter of M2,1 (F ) in terms of irreducible characters. In symbols: χST n (M2,1 (F )) =

X λ∈H(2,1,n)

χλ ⊗ χλ =

X

cµ χ µ .

µ

Denote by bxc the greatest integer less than or equal to x and dxe the least integer greater than or equal to x.

7

+3 < 42 > +2 < 32 > +3 < 3, 13 +2 < 23 > +5 < 2, 13 +3 < 14 >

Claim 1 m(n) = |H(k, l, n)| Proof. By using the Frobenius map (see [6], [12]) we can replace the irreducible Sn -character corresponding to λ with the Schur function sλ and we get: X m(n) =< sλ ◦ sλ , s(n) >= λ∈H(k,l,n)

X

< sλ ◦ sλ , s(n) >

λ∈H(k,l,n)

where denotes the usual Hall inner product on symmetric functions. Therefore, by using the basic properties of the Kronecker product of Schur functions, we get: m(n) =
=

λ∈H(k,l,n)

X

X

< sλ , sλ >=

1 = |H(k, l, n)|

λ∈H(k,l,n)

λ∈H(k,l,n)

t u Remark: An easy computation shows that n−2 X n−i 3n − 2 + = |H(2, 1, n)| = 2 2 i=3





c(n)





Claim 2 m(1n ) =the number of partitions of H(k, l, n) which are self conjugate. Proof. m(1n ) =

Similarly as in Claim 1 we get X

X

< sλ ◦sλ , s(1n ) >=

λ∈H(k,l,n)

< sλ ◦s(1n ) , sλ >=

λ∈H(k,l,n)

X

< sλ0 , sλ >

λ∈H(k,l,n)

where λ0 denotes the conjugate of λ and considering that (

< sλ0 , sλ >=

0 1

if λ0 6= λ if λ0 = λ t u

we are done.

Remark: c(1n ) is always equal to 1 except the case n = 2, when is zero. In fact by an easy computation, the only self conjugate partitions contained l m n n in H(2, 1, n) are the hooks ( 2 , 1b 2 c ) for n odd and the partition (a, 2, 1a−2 ) for n even with 2a = n. From now on we will use the reduced notation. 8

3.1

Computation of the coefficient for {n − 1, 1}

We want to establish the coefficient {n − 1, 1} as a polynomial in n. In reduced notation that amounts to determining the coefficient of < 1 > in the reduced product squares using equation < λ > ◦ < µ >=

X

< {λ/αβ} · {µ/αγ} · {β ◦ γ} >

α,β,γ

and remembering that {0, n} = −{n − 1, 1}. Therefore, since < 1 > ◦ < 1 >=< 2 > + < 12 > + < 1 > + < 0 > we are essentially interested in the sums of the inner squares of all the partitions of n in which the third part is 1 or 0. In reduced notation they are of the form < k1x >. Thus for n = 8 the partitions of interest are {8} +{521} +{414 } +{216 }

+{71} +{513 } +{32 12 } +{18 }

+{62} +{42 } +{3213 }

+{612 } +{431} +{315 }

+{53} +{4212 } +{22 14 }

(5)

In reduced notation, the single hooks are (not in the same order as above) + < 213 > + < 16 > +

+ < 312 > + < 31 > + < 3 > + < 214 > + < 212 > + < 21 > + < 2 > + < 17 > + < 15 > + < 14 > + < 13 > + < 12 > +

(6)

Notice that the above hooks can be arranged in groups as follows < 31 > < 21 > < 312 > < 214 > < 213 > < 212 > < 17 > < 16 > < 15 > < 14 >

(7) < 13 >

< 12 >

We do this to show that the various reduced inner squares can be divided into classes of hooks and each class can be enumerated and treated separately. Effectively we need to examine classes of the types < 1x >, < x > and < k1x > and decide on the permissible values of x, k. We need to know in each case how many times does the reduced < 1 > occur in each square. 9

First we consider the class < 1x >. Here we have the bound n−2 ≥ x ≥ 1. We have discarded x = n, n − 1 since {0, 1n } = 0 and {1, 1n−1 } = {1n } whose square can anly give {n}. So now what is the multiplicity of < 1 > in < 1x > ◦ < 1x >? Recall the general result < λ > ◦ < µ >=

X

< λ/αβ · µ/αγ · β ◦ γ >

(8)

α,β,γ

In case of λ = µ = 1x we will only be able to get < 1 > when α = 1x−1 and β = γ = 1, remembering that {1} ◦ {1} = {1}. Thus we obtain 1 for each permissible value of x. There are n − 2 permissible x and hence this class contributes n − 2 towards the final result. That result holds for n even or odd. Next consider the class < x >. We have already done x = 1 above and hence the lower bound on x is 2. Recalling that {0, n} = −{n − 1, 1} we will obtain, for n even, a < n > for x ≥ n2 which will cancel the < 1 >. Thus ≥ x ≥ 2 each permissible x contributes 1 leading to a we must have n−2 2 . For odd n we similarly obtain the total contribution from this class of n−4 2 bounds n−1 ≥ x ≥ 2 and obtain a total contribution of n−3 . 2 2 x Now consider the class < k1 > with k > 1. Putting λ, µ = k1x in (8) shows that there are two possibilities of obtaining < 1 > • α = k − 1, 1x and β = γ = 1 • α = k, 1x−1 and β = γ = 1 This suggests that each x contributes 2 but remember {0, n} = −{n − 1, 1}. Suppose x = n − 2k then (8) becomes < k1n−2k > ◦ < k1n−2k >=

X

< k1n−2k /αβ · k1n−2k /αγ · β ◦ γ >

(9)

α,β,γ

We can easily see that a single term, < n >, arises when α = 0 with β = γ = 1n−2k since < k · k · 1n−2k ◦ 1n−2k >=< k · k · n − 2k >⊃< n > and this term must be subtracted from the 2. Thus the rules: • Count 1 for each (k1x ) with x = n − 2k giving a contribution of

n−4 2

• Count 2 for each (k1x ) with n − 2k > x ≥ 1 giving a contribution of n−42 2

10

Adding the terms gives the coefficient of {n − 1, 1} for n even as (n − 2)2 2 Thus for n = 18 the coefficient of {17, 1} is 128. For n odd 1. Count 1 for each < 1x > with n − 2 ≥ x ≥ 1 giving n − 2. 2. Count 1 for each < x > with

n−1 2

≥ x ≥ 2 giving

n−3 . 2

3. Count 1 for each < k1x > with x = n − 2k giving

n2 −3 2

4. Count 2 for each < k1x > with n − 2k > x ≥ 1 and k > 1 giving n2 −8n+15 . 2 Adding the terms gives the coefficient of {n − 1, 1} for n odd as (n2 − 4n + 5) . 2 Thus for n = 17 the coefficient of {16, 1} is 113. In a similar way, we can compute the coefficients for {n − 2, 2} and {n − 2, 1 }. We omit the details for brevity. The final results are the following: 2

(

c({n−2,2}) =

3n2 −19n+34 2 3n2 −19n+32 2

if n is even if n is odd

for any n the coefficient of {n − 2, 12 } is n2 − 6n + 10. Remark The computational aspects of this paper were made using SCHUR, an interactive program for calculating the properties of Lie groups and symmetric functions by Brian G. Wybourne.

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References [1] V. Drensky, G.K.Genov, Multiplicities in the trace cocharacter sequence of two 4x4 matrices, to appear in Mediterr.J.Math. [2] D.E. Littlewood, The Kronecker product of symmetric group representations, J. London Math. Soc., (1) 31, 1956, 89-93. [3] D.E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, Clarendon Press, Oxford, 2nd ed., 1950. [4] D.E. Littlewood, Can. J. Math., 10, 1958a, 1-16. [5] D.E. Littlewood, Can. J. Math., 10, 1958b, 17-32. [6] I.J. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995. [7] F.D. Murnaghan, Am.J.Math., 60, 1938, 761-84. [8] C.Procesi, The invariant theory of nxn matrices, Advances in Math., 19, 1976, 306-381. [9] Y.P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Izv.Akad.Nauk.SSSR Ser. Math., 38, 1974, 337-360. [10] Y.P. Razmyslov, Trace identities and central polynomials in matrix superalgebras Mn,k , Mat.Sb. (N.S.)128 (170), 2, 1985, 194-215. [11] A.Regev, Sign trace identities, Linear and Multilinear Algebra, 21, 1987, 1-28. [12] J.B. Remmel, Computing Cocharacters of Sign Trace Identities, Linear and Multilinear Algebra, 23, 1988, 1-14. [13] T. Scharf, J.-Y. Thibon, B. Wybourne, Reduced notation, inner plethysms and the symmetric group, J. Phys. A: Math. Gen., 26, 1993, 7461-7478. [14] H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, NJ, 1946 .

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