Trace cocharacters and the Kronecker products of Schur functions

Journal of Algebra 260 (2003) 631–656 www.elsevier.com/locate/jalgebra

Trace cocharacters and the Kronecker products of Schur functions J.O. Carbonara,a,∗ L. Carini,b and J.B. Remmel c,1 a Department of Mathematics, State University of New York’s College, Buffalo, NY 14222-1095, USA b Dipartimento di Matematica ed Applicazioni, Universita di Palermo, Via Archirafi 34, 90123 Palermo, Italy c Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA

Received 1 November 2001 Communicated by Corrado de Concini

Abstract It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities
of the algebra Mr (F ) of r × r matrices over a field F of characteristic zero are given by TCr,n = λ∈Λr (n) χ λ ⊗ χ λ where χ λ ⊗ χ λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λr (n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ = (λ1
· · ·
λk ) with k
r. We study the behavior of the sequence of trace cocharacters TCr,n . In particular, we study the behavior of the coefficient of χ (ν,n−m) in TCr,n as a function of n where ν = (ν1
· · ·
νk ) is some fixed partition of m and n − m  νk . Our main result shows that such coefficients always grow as a polynomial in n of degree r − 1.  2003 Elsevier Science (USA). All rights reserved.

1. Introduction The theory of trace identities, developed independently by Procesi and Razmyslov [P1,R], has proved to be a powerful tool in the study of identities of the algebra Mr (F ) of r × r matrices over a field F of characteristic zero. One can show that the group algebra F Sn of the symmetric group can be identified with multilinear trace polynomials. Then one can use the classical work of Schur and Weyl on the polynomial representations of * Corresponding author.

E-mail addresses: [email protected] (J.O. Carbonara), [email protected] (L. Carini). 1 Partially supported by ARO grant DAAD19-01-1-0724.

0021-8693/03/$ – see front matter  2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-8693(03)00013-9

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the general linear algebra gl(r, C) to show that the trace cocharacters χnT of Mr (F ) equals TCn,r where 

TC n,r =

χ λ ⊗ χ λ.

(1)

λ∈Λr (n)

Here χ λ ⊗ χ λ denotes the Kronecker or inner product of the irreducible Sn -character χ λ with itself and Λr (n) denotes the set of partitions of n, λ = (λ1
· · ·
λk ), with at most r non-zero parts. Trace identities help in describing the ordinary identities of matrices. Indeed in [F1,F2], Formanek showed that the (ordinary) cocharacters which describe the identities of r × r matrices, are almost identical with the above trace cocharacters TCr,n . It follows from the basic properties of Kronecker products that 

χλ ⊗ χλ =

λ∈Λr (n)



  mµ Mr (F ) χ µ

(2)

µ∈Λr 2 (n)

where mµ (Mr (F )) are non-negative integers. That is, one can construct a representation of the symmetric group Sn whose character is given by the left-hand side of (2) so that mµ (Mr (F )) represents the number of times the irreducible representation corresponding to the partition µ occurs in such a representation. Via the Frobenius map which makes the center of the group algebra of Sn onto the space of homogeneous symmetric functions of degree n, one can rephrase (2) in terms of symmetric functions as  λ∈Λr (n)

sλ ⊗ sλ =



  mµ Mr (F ) sµ

(3)

µ∈Λr 2 (n)

where sλ denotes the Schur function associated with the partition λ and ⊗ denotes the Kronecker product of Schur functions. Thus the coefficients mµ (Mr (F )) arise naturally in P.I. theory, invariant theory, the representation theory of the symmetric group, and the theory of symmetric functions. A central problem is to find formulas or algorithms to compute the coefficients mµ (Mr (F )). In general, relatively little is known about these coefficients. In the case r = 2, Carini and Regev in [CR] proved explicit formulas for the coefficients mµ (M2 (F )). The methods of [CR] are based on results due to Procesi [P2] (see also [DR]) and, in part, on the combinatorial description of the expansion of χ (a,b) ⊗ χ (c,d) obtained by Remmel and Whitehead in [R-Whd]. For r  3, Berele [B] has developed some formulas which allow one to obtain information about the asymptotic behavior of the coefficients m(µ1 ,µ2 ) (M3 (F )). There seems to be no substantial results about the coefficients mµ (Mr (F )) for r  4. The goal of this paper is to study the behavior of the coefficients mµ (Mr (F )) for certain classes of µ. That is, suppose that we fix a partition of m, ν = (ν1
· · ·
νk ) and n is large enough so that n − m  νk . We let |ν| = m denote the size of ν and we let (ν, n − |ν|) denote the partition (ν1
· · ·
νk , n − |ν|). We shall study the behavior of the coefficients m(ν,n−|ν|) (Mr (F )) as n → ∞. Our main result is the following:

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ν (x), . . . , P ν For any fixed partition ν of m, there are rational polynomials Pr,0 r,ur −1 (x) of degree r − 1 with the same leading term and a constant cµ,r such that, for all n  cµ,r , ν m(ν,n−m) (Mr (F )) = Pr,n mod (ur ) (n).

Here the sequence u1 , u2 , . . . is defined by induction as u1 = 1 and un = LCM(un−1 , n) where LCM(a, b) is the least common multiple of the integers a and b. Thus there are ur polynomials of degree r − 1 such that for large n the coefficient m(ν,n−m) (Mr (F )) can be found by evaluating one of these polynomials at n. Hence the coefficients m(ν,n−m) (Mr (F )) grow like a polynomial of degree r − 1. We note in the case of r = 2, this result easily follows from the work of Carini and Regev [CR]. However this result is new for all r  3. Our proof proceeds by induction on r and is based on the combinatorial properties of the expansion of the Kronecker product of two Schur functions. One of the key steps uses an algorithm of Garsia and Remmel [GR] which reduces the computation of Kronecker products of Schur functions to the computations of sums and differences of products of skew Schur functions. The outline of this paper is as follows. In Section 2 we shall state the basic formulas and algorithms that are needed to expand the product of two Schur functions and the Kronecker product of two Schur functions as a sum of Schur functions. These algorithms are needed to carry out the computation of the coefficients mµ (Mr (F )). In Section 3 we shall apply those formulas to prove our main result. In Section 4, we shall provide some tables of the coefficients m(ν,n−|ν|) (Mr (F )) and the polynomials Pkν (x) for a few small values of r and partitions ν.

2. Basic formulas and algorithms
Given the partition λ = (λ1 , λ2 , . . . , λk ) where 0 < λ1
λ2
· · ·
λk and λj = n, we let Fλ denote the Ferrers’ diagram of λ, i.e. Fλ is the set of left-justified squares or boxes with λ1 squares in the top row, λ2 squares in the second row, etc. For example see Fig. 1. For the sake of convenience, we will often refer to the diagram Fλ simply by λ. We let |λ| = n denote the size of λ and we let the length of λ, l(λ), denote the number of parts of λ. We let λ denote the conjugate partition of λ, i.e. the partition whose parts are the heights of the columns of Fλ . Given two partitions λ = (λ1 , . . . , λk ) and µ = (µ1 , . . . , µl ), we write λ
µ if and only if k
l and λk−p
µl−p for 0
p
k − 1. If λ
µ, we let |λ/µ| = |λ| − |µ|. We let Fµ/λ denote the Ferrers’ diagram of the skew shape µ/λ where Fµ/λ is the diagram that results by removing the boxes corresponding to Fλ from the diagram Fµ . For example, F(2,3,3,4)/(2,2,3) consists of the lighter shaded boxes in Fig. 2.

F(2,3,3,4) =

Fig. 1.

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J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

Fig. 2.

Fig. 3.


Let λ n and α = (α1 , α2 , . . . , αk ) be a sequence of positive integers such that αi = n. Define a decomposition of λ of type α, denoted by D1 + D2 + · · · + Dk = λ, as a sequence of shapes φ = λ0 ⊂ λ1 ⊂ λ2 ⊂ · · · ⊂ λk = λ with λi /λi−1 a skew shape, Di = λi /λi−1 , and |Di | = αi . For example, for k = 2, λ = (2, 3), α1 = 2, α2 = 3, the two decompositions of λ of type α are pictured in Fig. 3 where the darker portion corresponds to D1 and the rest corresponds to D2 . A column strict tableau T of shape µ/λ is a filling of Fµ/λ with positive integers so that the numbers weakly increase from left to right in each row and strictly increase from bottom to top in each column. T is said to be standard if the entries of T are precisely the numbers 1, 2, . . . , n where n equals |µ/λ|. We let CS(µ/λ) and ST(µ/λ) denote the set of all column strict tableaux and standard tableaux of shape µ/λ respectively. Given T ∈ CS(µ/λ), the weight of T , denoted by ω(T ), is the monomial obtained by replacing each i in T by xi and taking the product over all boxes. For example,

if

T=

,

then ω(T ) = x12 x23 x3 .

This given, the skew Schur function sµ/λ is defined by sµ/λ (x1 , x2 , . . .) =



ω(T ).

(4)

T ∈CS(µ/λ)

The special case of (4) where λ is the empty diagram, i.e. λ = ∅, defines the usual Schur function sµ . For emphasis, we shall often refer to those shapes which arise directly from partitions µ as straight shapes so as to distinguish them among the general class of skew shapes. One can also show that for all λ, µ, and ν such that |λ| + |ν| = |µ|, sλ sν , sµ  = sν , sλ/µ ,

(5)

see [Mac]. Here  ,  denotes the usual scalar product of the space of homogeneous symmetric functions of degree n under which the Schur functions {sλ : |λ| = n} form a

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

635

F(1,2)∗(2,3) =

Fig. 4.

Fig. 5.

complete orthonormal system. For an integer n, the Schur function indexed by the partition (n) is also called the nth homogeneous symmetric function and will be denoted by hn . Thus, hn = s(n) . Given a partition λ = (λ1 , . . . , λk ), we let hλ = hλ1 · · · hλk . Next we shall describe a rule due to Remmel and Whitney [R-W] for expanding the product of skew Schur functions as a sum of Schur functions. Given partitions λ and µ, let λ ∗ µ denote the skew diagram that results from Fλ and Fµ by placing Fλ on top of Fµ so that the start of the top row of Fλ is just below the end of the bottom row of Fµ . For example see Fig. 4. It is easy to see that sλ∗µ = sλ sµ so that expanding the product of two Schur functions as a sum of Schur functions is just a special case of expanding an arbitrary skew Schur function as a sum of Schur functions. Moreover, it should be clear that the problem of expanding an arbitrary product of Schur functions or skew Schur functions corresponds to expanding a single skew Schur function as a sum of Schur functions. For example, s(2,3) · s(1,2) · s(4,4,4)/(1,2) is equal to the skew Schur function whose Ferrers diagram is pictured in Fig. 5. Such expansions can be computed via the following version of the Littlewood– Richardson rule due to Remmel and Whitney (see [R-W]). Skew Schur function
expansion rule. ν To compute sλ/µ = ν cλ/µ sν : (1) Form the reverse lexicographic filling of λ/µ, rl(λ/µ), which is the filling of Fλ/µ which starts at the bottom right corner of Fλ/µ and fills in the integers 1, 2, . . . , n = |λ/µ| in order from right to left and bottom to top. For example see Fig. 6. (2) We say a standard tableau T is (λ/µ)-compatible if (a) whenever i + 1 is immediately to the left of i in rl(λ/µ), then in T , i + 1 occurs to the southeast of i in the sense that the cell of T which contains i + 1 is strictly to the right and weakly below the cell of T which contains i;

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J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

rl(F(1233)/(12) ) =

rl(F(4,4,4,4)/(12) ) =

Fig. 6.

(b) whenever y is immediately above x in rl(λ/µ), then in T , y occurs to the northwest of x in the sense that the cell of T which contains y is strictly above and weakly to the left of the cell of T which contains x. ν is the number of (λ/µ)-compatible tableaux of shape ν. Then cλ/µ

It is good to visualize the condition i +1 southeast of i and y northwest of x respectively by the patterns ii+1 yx . Thus for the example λ/µ = (1, 2, 3, 3)/(1, 2) pictured above, conditions (a) and (b) may be summarized by the patterns 23 , 45 ,

21 , 43 , 65 .

We note that the collection of (λ/µ)-compatible tableaux can easily be constructed by adding squares labeled 1, 2, . . . , n in succession, always maintaining standardness and each time obeying conditions (a) and (b). In our example, one is naturally led to the tree in Fig. 7 for constructing the (λ/µ)-compatible tableaux. Having constructed the tree, one can easily read off the expansion of sλ/µ as sλ/µ =

 T : (λ/µ)-compatible

Fig. 7.

ssh(T ) .

(6)

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

637

Thus for our example, s(1,2,3,3)/(1,2) = s(12 ,22 ) + s(23 ) + s(13 ,3) + 2s(1,2,3) + s(32 ) . We also have the following identity for Schur functions, called the Jacobi–Trudi identity: sλ = dethλj −i+j 1
i,j
l(λ)

(7)

where h0 = 1 and for r < 0, hr = 0. Proof of this theorem can be found in [Mac]. We now state some properties of the Kronecker product. Suppose that λ and µ are partitions of n and sλ ⊗ sµ =



gλµν sν .

(8)

ν

Then we have the following: hn ⊗ sλ = sλ ,

(9) where λ denotes the conjugate of λ,

s(1n ) ⊗ sλ = sλ

sλ ⊗ sµ = sµ ⊗ sλ = sλ ⊗ sµ = sµ ⊗ sλ , (P + Q) ⊗ R = P ⊗ R + Q ⊗ R, gλµν = gλνµ = gνλµ = gνµλ = gµλν = gµνλ .

(10) (11) (12) (13)

Here in (12), P , Q, and R are arbitrary symmetric functions. We note that (9) through (12) can be easily established by the definition of Kronecker product. A proof of (13) can be found in [Lw]. Littlewood [Lw] proved that (sα sβ ) ⊗ sλ =

 

cγ δλ (sα ⊗ sγ ) (sβ ⊗ sδ )

(14)

γ |α| δ |β|

where γ , δ and λ are straight shapes and cγ δλ is the Littlewood–Richardson coefficient, i.e. cγ δλ = sγ sδ , sλ . Garsia and Remmel [GR] then used (14) to prove the following: (sH · sK ) ⊗ sD =



(sH ⊗ sD1 ) · (sK ⊗ sD2 )

(15)

D1 +D2 =D |D1 |=|H | |D2 |=|K|

where H , K, and D are skew shapes and the sum runs over all decompositions of the skew shape D. In particular, one can easily establish by induction from (15) that

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J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656



(ha1 · · · hak ) ⊗ sD =

sD1 · · · sDk ,

(16)

D1 +···+Dk =D |Di |=ai

where the sum runs over all decompositions of D of length k such that |Di | = ai for all i.

3. Main result As stated in the introduction, our main interest in this paper is to study the behavior of 

sλ ⊗ sλ ,

(17)

λ n l(λ)
r

where the symbol λ n denotes that λ is a partition of n. Now fix a partition δ = (δ1
· · ·
δk ). For any n  |δ| + δk , let Crδ (n) =



sλ ⊗ sλ , s(δ,n−|δ|) .

(18)

λ n l(λ)
r

Define {ur }r1 inductively by u1 = 1 and ur = LCM(ur−1 , r) where LCM(s, r) denotes the least common multiple of s and r. Our main goal is to prove the following theorem. δ (n), . . . , P δ Theorem 1. There are rational polynomials Pr,0 r,ur −1 (n) of degree r − 1 for any fixed partition δ, and a constant cδ,r such that for all n  cδ,r δ Crδ (n) = Pr,n mod (ur ) (n).

(19)

δ (n), . . . , Moreover, the coefficient of degree r − 1 is the same in all the polynomials Pr,0 δ (n). Pr,u r −1

Proof. If we use the Jacobi–Trudi identity s(δ,n−|δ|) = sν = dethνi +j −i  to expand s(δ,n−|δ|) as a signed sum of homogeneous symmetric functions and apply identity (12), it is easy to see that Crδ (n) can be expressed as a signed sum of terms of the form γ

Er (n) =



sλ ⊗ sλ , hγ hn−|γ | 

(20)

λ n l(λ)
r

where γ is a partition such that |γ |
|δ|. Thus we can prove Theorem 1 if we can show that for all partitions γ = (γ1 , . . . , γl(γ )) and all n  |γ | + γl(γ ) there is a constant eγ ,r and γ γ polynomials Qr,0 (x), . . . , Qr,ur −1 (x) of degree r − 1 with the same leading term such that for all n  eγ ,r γ

γ

Er (n) = Qr,n mod (ur ) (n).

(21)

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

Fν =

,

  ν = 3, 4, 64 ,   ν  = 42 , 5, 63 ,

Fλ =

,

λ = (2, 4, 5),   λ = 1, 22 , 32 ,

Fν⊕λ =

639

;

ν ⊕ λ = (3, 4, 6, 8, 10, 11);   (ν ⊕ λ) = 1, 22 , 32 , 42 , 5, 63 .

Fig. 8.

We shall prove this by induction on r. In fact, we need to prove something more general. That is, suppose ν is a partition whose smallest column height is greater that r. Thus ν  = ((r + 1)br+1 (r + 2)br+2 · · · s bs ) for some s  r + 1. Next let λ be a partition such that l(λ)
r so that λ must be of the form λ = (1a1 2a2 · · · r ar ). We can then define ν ⊕ λ by setting (ν ⊕ λ) = (1a1 · · · r ar (r + 1)br+1 · · · s bs ). Thus the Ferrers diagram of ν ⊕ λ is the result of attaching the Ferrers diagram of λ to the right of the Ferrers diagram of ν as pictured in Fig. 8. Now fix ν such that ν  = ((r + 1)br+1 (r + 2)br+2 · · · s bs ). Then for t
r and n  |γ | + γk , define γ ,ν

Et



(n) =

sν⊕λ ⊗ sν⊕λ , hγ hn−|γ | .

(22)

λ n−|ν| l(λ)
t

We shall prove the following lemma. Lemma 1. With the notation above, for each 1
t
r, there exist rational polynomials γ ,ν γ ,ν Qt,0 (x), . . . , Qt,ut −1 (x) of degree t − 1 and a constant eγ ,ν,t such that for each n  eγ ,ν,t γ ,ν

γ ,ν

Et (n) = Qt,n mod (ut ) (n).

(23) γ ,ν

Moreover, the coefficient of degree r − 1 is the same in all the polynomials Qt,0 (x), . . . , γ ,ν Qt,ut −1 (x). Proof. We proceed by induction on t. First consider a term in (22). Using the symmetry of the Kronecker coefficients, the Garsia–Remmel rule, and (5), one can show that Gγ ,ν⊕λ = sν⊕λ ⊗ sν⊕λ , hγ hn−|γ |  = hγ hn−|γ | ⊗ sν⊕λ , sν⊕λ   hγ ⊗ sν⊕λ/µ , sν⊕λ/µ . = µ⊆ν⊕λ |ν⊕λ/µ|=|γ |

(24)

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J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

δ=

→ ↓ red3 (δ) =

Fig. 9.

Now for any partition δ with |δ|  |γ |, consider the sum Gγ ,δ =



hγ ⊗ sδ/µ , sδ/µ .

(25)

µ⊆δ |δ/µ|=|γ |

Clearly Gγ ,δ depends only on the skew shapes of size |γ | that we can remove from δ that results in a normal partition. Now if the Ferrers diagram of δ has ω columns of height k where ω > |γ |, then the first ω − |γ | of height k, reading from left to right, can never be involved in one of these skew shapes. Thus removing the first ω − |γ | of height k from δ will leave us with a partition δ such that 



hγ ⊗ sδ/µ , sδ/µ  =

µ⊆δ |δ/µ|=|γ |

hγ ⊗ sδ/µ , sδ/µ 

(26)

µ⊆δ |δ/µ|=|γ |

since both sides of (26) involve exactly the same skew shapes. This leads us to define the k-reduction of a shape δ, redk (δ), by     redk (δ) = 1min(k,a1 ) , . . . , nmin(k,ak ) where δ  = 1a1 2a2 · · · nan . That is, redk (δ) is the result of removing the first wt − k columns of height t in δ from δ for each t such that δ has wt > k columns of height t. For example, if k = 3 and δ = (4, 6, 11), then δ  = 15 22 34 so that (red3 (δ)) = 13 22 33 and red3 (δ) = (3, 5, 8). Thus Fred3 (δ) simply results from δ by replacing any sequence of more than 3 columns of height j in Fδ by exactly 3 columns of height j . It is then easy to see from Fig. 9 that there is a one to one correspondence between {δ/µ: |δ/µ| = 3} and {red3 (δ)/µ: | red3 (δ)/µ| = 3} since no element of a skew shape δ/µ with |δ/µ| = 3 can involve any square of the columns that we removed from δ to get red3 (δ). It follows that Gγ ,δ = Gγ ,red|γ | (δ) .

(27)

Thus if γ k, then γ ,ν

Et

(n) =

 λ n−|ν| l(λ)
t

Gγ ,redk (ν⊕λ)

(28)

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

641

where 

Gγ ,redk (ν⊕λ) =

hγ ⊗ sredk (ν⊕λ)/µ, sredk (ν⊕λ)µ .

(29)

µ⊆redk (ν⊕λ) | redk (ν⊕λ)/µ|=k

Now for t = 1, there is only one term in the outer sum, namely, λ = (n − |ν|). However as soon as n − |ν|  k, then redk (ν ⊕ (n − |ν|)) = redk (ν ⊕ (k)) so that for all n  k + |ν| γ ,ν

E1 (n) = Gγ ,redk (ν⊕(k)) . γ ,ν

(30) γ ,ν

Thus E1 (n) is eventually constant and hence Lemma 1 holds for E1 (n) where eγ ,ν,1 = k + |ν|. Next suppose that r  2 and γ k. Then 

γ ,ν

E2 (n + |ν|) =

n

Gγ ,redk (ν⊕(a,a+b)).

(31)

0
a
2 b=n−2a

Thus if 2m  3k, then γ ,ν

γ ,ν

E2 (2m + 2 + |ν|) − E2 (2m + |ν|)    = Gγ ,redk (ν⊕(m+1,m+1)) + Gγ ,redk (ν⊕(a,a+b+2)) − Gγ ,redk (ν⊕(a,a+b)) . (32) 0
a
m b=2m−2a

Now it is easy to see that if b  k, then     redk ν ⊕ (a, a + b + 2) = redk ν ⊕ (a, a + b) .

(33)

Similarly if b = 2m − 2a < k, then a  k, since we are assuming 2m  3k, and     redk ν ⊕ (a, a + b) = redk ν ⊕ (k, k + b) .

(34)

    redk ν ⊕ (m + 1, m + 1) = redk ν ⊕ (k, k) .

(35)

Finally, m  k; so that

Thus combining (31)–(34) we can show that γ ,ν

γ ,ν

E2 (2m + 2 + |ν|) − E2 (2m + |ν|)    = Gγ ,redk (ν⊕(k,k)) + Gγ ,redk (ν⊕(k,k+2c+2)) − Gγ ,redk (ν⊕(k,k+2c)) k

0
c


=G

2

γ ,redk (ν⊕(k,k+2k/2+2))

= Gγ ,redk (ν⊕(k,2k)).

A similar computation will show that

(36)

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J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656 γ ,ν

γ ,ν

E2 (2m + 3 + |ν|) − E2 (2m + 1 + |ν|)    = Gγ ,redk (ν⊕(m+1,m+2)) + Gγ ,redk (ν⊕(a,a+b+3)) − Gγ ,redk (ν⊕(a,a+b+1)) 0
a
m b=2(m−a)

= Gγ ,redk (ν⊕(k,k+1)) +

  k

0
c


=G

γ ,redk (ν⊕(k,k+2 2k +3))

Gγ ,redk (ν⊕(k,k+2c+3)) − Gγ ,redk (ν⊕(k,k+2c+1))



2

= Gγ ,redk (ν⊕(k,2k)),

(37)

where for any rational number q, q denotes the greatest integer less than or equal to q. It γ ,ν γ ,ν is clear that (36) and (37) imply that the functions E2 (2m + |ν|) and E2 (2m + 1 + |ν|) are eventually linear functions of m with the same leading coefficients. Formally, we let mo be the least m such that 2mo  3k. It then follows from (36) and (37) that if n  2mo + |ν|, then   n−|ν|

 γ ,ν 2 − mo W + E2 (2mo + |ν|)   n−1−|ν| γ ,ν − mo W + E2 (2mo + 1 + |ν|) 2

γ ,ν E2 (n) =

(n − |ν| even), (n − |ν| odd),

(38)

where W = Gγ ,redk (ν⊕(k,2k)). Thus Lemma 1 holds for t = 2 for any ν such that ν  is of γ ,¯ν the form (3b3 4b4 · · · s bs ). Now suppose j < r and that Lemma 1 holds for Ej (n) for all ¯

¯

¯

ν¯ where ν¯  is of the form ((j + 1)bj+1 (j + 2)bj+2 · · · s¯bs¯ ). Since we can partition the set of partitions λ with l(λ)
j +1 by the number of columns of size j + 1 in Fλ , we have that γ ,ν Ej +1 (n) =

(n−|ν|)/j  +1

γ ,ν⊕(p j+1 )

Ej

(n).

(39)

p=0

However, if p  k then by (26) γ ,ν⊕(p j+1 )

Ej



(n) =

Gγ ,(ν⊕p

j+1 )⊕λ

λ n−|ν|−p(j +1) l(λ)
j γ ,ν⊕(k j+1 ) 

= Ej



=

Gγ ,(ν⊕k

j+1 )⊕λ

λ n−|ν|−p(j +1) l(λ)
j

 n − (p − k)(j + 1) .

(40)

Thus

γ ,ν Ej +1 (n)

=

k−1  p=0

 n−|ν| 

γ ,ν⊕(p j+1 ) Ej (n) +

j+1 

p=k

γ ,ν⊕(k j+1 ) 

Ej

 n − (p − k)(j + 1) .

(41)

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

643

By induction, there are polynomials of degree j − 1 with the same leading term, γ ,ν⊕(p j+1 )

Qj,0

γ ,ν⊕(p j+1 )

(x), . . . , Qj,uj −1

(x)

and constants eγ ,ν⊕pj+1 ,j for p = 0, . . . , k − 1 such that γ ,ν⊕p j+1

Ej

γ ,ν⊕p j+1

(n) = Qj,n mod (uj ) (n)

for n  eγ ,ν⊕pj+1 ,j .

(42)

We note, however, that there is a parity mismatch in the sum  n−|ν|  j+1 

γ ,ν⊕(k j+1 ) 

n − (p − k)(j + 1)

Ej



p=k γ ,ν⊕(k j+1 )

in that the Ej (n − (p − k)(j + 1)) are polynomials in n − (p − k)(j + 1) mod uj , but we are subtracting multiples of j + 1 in the arguments. Thus the most natural thing to do is to consider the sum relative to n mod uj +1 where uj +1 = LCM(j + 1, uj ). To this end, we let a and b be such that a(j + 1) = b(uj ) = uj +1 . Also let rγ ,ν,j +1 = max{eγ ,ν⊕(pj+1 ) : p = 0, . . . , k − 1} and γ ,ν

Rj +1,s (x) =

k−1 

γ ,ν⊕(p j+1 )

Qj,s mod (uj ) (x) for s = 0, . . . , uj +1 − 1.

(43)

p=0 γ ,ν

Then it is easy to see that each Rj +1,s (x) is a polynomial of degree j − 1 and for all n  rγ ,ν,j +1 k−1 

γ ,ν⊕p j+1

Ej

γ ,ν

(n) = Rj +1,n mod (uj ) (n).

(44)

p=0

Note that the numbers 0, . . . , uj +1 − 1 can be divided up according to their value mod uj as  



0, uj , . . . , (b − 1)uj ,

1, 1 + uj , . . . , 1 + (b − 1)uj , .. .

uj − 1, uj − 1 + uj , . . . , uj − 1 + (b − 1)uj .

(45)

644

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656 γ ,ν⊕p j+1

γ ,ν⊕p j+1

Thus if we let Sj +1,i+cuj (x) = Rj +1 is easy to see that k−1 

γ ,ν⊕p j+1

Ej

(x) for 0
i
uj − 1 and 0
c
b − 1, then it

γ ,ν

(n) = Sj +1,n mod (uj+1 ) (n).

(46)

p=0

Next consider the sum  n−|ν| 

B=

j+1 

γ ,ν⊕(k j+1 ) 

Ej

 n − (p − k)(j + 1) .

p=0

Note that the numbers {n − s(j + 1): s = 0, . . . ,  n−|ν| j +1 } can be partitioned according to their values mod uj +1 as  0, a(j + 1), 2a(j + 1), . . . ,  j + 1, j + 1 + a(j + 1), j + 1 + 2a(j + 1), . . . , .. .

 (a − 1)(j + 1), (a − 1)(j + 1) + a(j + 1), (a − 1)(j + 1) + 2a(j + 1), . . . . Let eγ ,ν,j +1 = rγ ,ν,j +1 + uj +1 . Now fix n  eγ ,ν,j +1 and suppose that for t = 0, . . . , a − 1, n − t (j + 1) = it + ct uj

where 0
it
uj − 1.

(47)

Note that i0 , . . . , ia−1 must be distinct. That is, if iu = iω for some 0
u < ω
a − 1, then (n − u(j + 1)) − (n − ω(j + 1)) = (ω − u)(j + 1) ∼ = 0 mod (uj ), which would violate the fact that a(j + 1) = LCM(j + 1, uj ). Next for each t = 0, . . . , a − 1, let ct = x t b + y t

where 0
yt < b.

(48)

Thus for fixed t with 0
t
a − 1, the positive elements in {n − (p − k)(j + 1): k
p
 n−|ν| j +1 } of the form n − t (j + 1) − suj +1 = n − t (j + 1) − sbuj are 

n − t (j + 1), n − t (j + 1) − buj , . . . , n − t (j + 1) − xt buj = {it + yt uj , it + yt uj + buj , . . . , it + yt uj + xt buj }.

Note that ct =

n − t (j + 1) − it , uj

yt =

n − t (j + 1) − it mod b, uj

and



J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

xt =

645

n − t (j + 1) − it − yt uj . buj

Thus  n−|ν| 

B =

j+1 

γ ,ν⊕(k j+1 ) 

Ej

n − (p − k)(j + 1)



p=k

=

a−1 

n−t (j+1)−it −yt uj buj



t =0

γ ,ν⊕(k j+1 )

Ej

(it + yt uj + sbuj ),

(49)

s=0

γ ,ν⊕k j+1

where we interpret Ej (m) = 0 if m
|ν| + k(j + 1). For each t = 0, . . . , a − 1, let rt be the smallest r such that it + yt uj + rbuj  eγ ,ν,j +1 . Thus

B =

a−1 

n−t (j+1)−it −yt uj buj



t =0

+

γ ,ν⊕(k j+1 )

Ej

(it + yt uj + sbuj )

s=rt

s−1 r t −1 

γ ,ν⊕(k j+1 )

Ej

(it + yt uj + sbuj ).

(50)

t =0 s=0

The second double sum in (50) is just some constant cl that depends on l = n mod (uj +1 ) γ ,ν⊕(k j+1 )

for n  eγ ,ν,j +1 . By our induction hypothesis, each term, Ej

γ ,ν⊕(k j+1 )

in the first double sum in (50) can be replaced by Qj,it

B = cn mod (uj+1 ) +

a−1  t =0

n−t (j+1)−it −yt uj buj



γ ,ν⊕(k j+1 )

Qj,it

(it + yt uj + sbuj ),

(it + yt uj + sbuj ). Thus

(it + yt uj + sbuj ).

(51)

s=rt

It is well known that for any integer u  0, there is a rational polynomial pu (x) of degree u + 1 such that r 

l u = pu (r).

(52)

l=0

Thus suppose γ ,ν⊕(k j+1 )

Qj,it

(x) = f0,j,it + f1,j,it x + · · · + fj −1,j,it x j −1 .

(53)

646

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

Then clearly there are rational numbers g0,j,it , . . . , gj −1,j,it such that γ ,ν⊕(k j +1)

Qj,it

(it + yt uj + sbuj ) = g0,j,it + g1,j,it s + · · · + gj −1,j,it s j −1 .

(54)

Moreover, our assumption is that the coefficient fj −1,j,it are all equal for t = 0, . . . , a − 1, so that the coefficient gj −1,j,it are all equal for t = 0, . . . , a − 1. It follows that for any t = 0, . . . , a − 1 n−t (j+1)−it −yt uj buj



γ ,ν⊕(k j+1 )

Qj,it

(it + yt uj + sbuj )

s=rt n−t (j+1)−it −yt uj buj



=

g0,j,it + g1,j,it s + · · · + gj −1,j,it s j −1

s=rt

=

j −1  z=0





 n − t (j + 1) − it − yt uj gz,j,it pz − pz (rt − 1) buj

(55)

is just some polynomial of degree j in n. Moreover, it is clear that the leading term of this polynomial is the same for all t = 0, . . . , a − 1. Combining (41), (51), and (55), we get γ ,ν γ ,ν that there exists rational polynomials Qj +1,0 (x), . . . , Qj +1,uj+1 −1 (x) of degree j with the same leading terms such that for all n  eγ ,ν, j +1 γ ,ν

γ ,ν

Ej +1 (n) = Qj +1,n mod (uj+1 ) (n)

(56)

as desired.

4. Examples and tables In this section, we shall give some explicit examples of Theorem 1 and Lemma 1. First we will shall consider the polynomials that arise in Lemma 1. In the case, where γ is the empty partition {}, then {},ν

Et

(n) =



sν⊕λ ⊗ sν⊕λ , hn 

(57)

λ n−|ν| l(λ)
t

=

 λ n−|ν| l(λ)
t

hn ⊗ sν⊕λ , sν⊕λ  =

 λ n−|ν| l(λ)
t

sν⊕λ , sν⊕λ  =

 λ n−|ν| l(λ)
t

1.

(58)

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

647

{},ν

Thus Et (n) is just the number of partitions of n − |ν| with less than or equal to t parts. If we let pt (n) denote the number of partitions of n with t or fewer parts, then there is a well-known generating function for such partitions, namely, ∞ 

pt (n)q = n

n=0

t i=1

1 . 1 − qi

{},{}

Table 1, labeled γ = {}, gives the values of Et (n) for t
4 and n
50 and the {},{} corresponding values of the polynomials Qt,j (x). In each of these tables, there is a line {},ν

s
x under the column headed by Et

{},ν

and its corresponding polynomial Qt,j (x). The {},ν

s in this case indicates the lowest value of x for which the polynomials Qt,j (x) give the {},ν

correct value for Et . One will see in this and subsequent tables that this value of s is much lower that the bound given in the proof of Lemma 1. In Table 2 we consider the next simplest case, namely, when γ = {1}. In this case, {1},ν

Et

(n) =

 λ n−|ν| l(λ)
t

=



sν⊕λ ⊗ sν⊕λ , h1 hn−1  =



h1 hn−1 ⊗ sν⊕λ , sν⊕λ 

λ n−|ν| l(λ)
t



s(1) sβ , sν⊕λ .

(59)

λ n−|ν| β⊆ν⊕λ l(λ)
t |(ν⊕λ)/β|=1

However it is easy to see from the skew Schur function multiplication rule that 

s(1) sβ , sν⊕λ 

β⊆ν⊕λ |(nu⊕λ)/β|=1

is just the number of squares of Fν⊕λ such that if we remove that square, we get a diagram of a partition. Such squares are called corner squares. For any partition µ, we let c(µ) denote the number of corner squares of µ. For example, it is easy to see from our picture of F(2,3,3,4) in Section 1 that c(2, 3, 3, 4) = 3. It follows that {1},ν

Et

(n) =



c(ν ⊕ λ).

λ n−|ν| l(λ)
t {1},{1s }

In Table 2 labeled γ = {1}, we list the values of Et (n) and the corresponding polyno{1},{1s } (n) for t
3, n
50 and s
4. These are the polynomials that are required mials Qt,j {1},{}

{1},{}

to compute E4 (n) and its corresponding polynomial Qt,j (n) which we also list. γ ,ν γ ,ν Finally will list the values of the Et (n) and the corresponding polynomials Qt,j (x) that are needed to compute the values of the polynomials Ctδ (x) for |δ|
3 and t
3. These final polynomials are given in Table 8.

648

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

Table 1 γ = {} γ , {}

γ , {}

γ , {}

γ , {}

n

E1

E2

E3

E4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26

1 1 2 3 4 5 7 8 10 12 14 16 19 21 24 27 30 33 37 40 44 48 52 56 61 65 70 75 80 85 91 96 102 108 114 120 127 133 140 147 154 161 169 176 184 192 200 208 217 225 234

1 1 2 3 5 6 9 11 15 18 23 27 34 39 47 54 64 72 84 94 108 120 136 150 169 185 206 225 249 270 297 321 351 378 411 441 478 511 551 588 632 672 720 764 816 864 920 972 1033 1089 1154

γ , {}

γ ,{}

E1 1
x

E2 1
x 1 x + = Q1 2 2 x 1 + = Q0 2

1 = Q0

γ ,{}

E3

1
x 5 12 2 3 3 4 2 3 5 12

+ + + + +

1+

x 2 x 2 x 2 x 2 x 2 x 2

+ + + + + +

x2 12 x2 12 x2 12 x2 12 x2 12 x2 12

= Q1 = Q2 = Q3 = Q4 = Q5 = Q0

γ ,{}

E4 1
x 65 7x 5x 2 + + 144 16 48 19 x 5x 2 + + 36 2 48 7x 5x 2 9 + + 16 16 48 8 x 5x 2 + + 9 2 48 49 7x 5x 2 + + 144 16 48 5x 2 3 x + + 4 2 48 65 7x 5x 2 + + 144 16 48 7 x 5x 2 + + 9 2 48 9 7x 5x 2 + + 16 16 48 23 x 5x 2 + + 36 2 48 49 7x 5 x2 + + 144 16 48 x 5x 2 1+ + 2 48

+ + + + + + + + + + + +

x3 144 x3 144 x3 144 x3 144 x3 144 x3 144 x3 144 x3 144 x3 144 x3 144 x3 144 x3 144

= Q1 = Q2 = Q3 = Q4 = Q5 = Q6 = Q7 = Q8 = Q9 = Q10 = Q11 = Q0

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

649

Table 2 γ = {1} γ ,{}

n E1 0 0 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 14 1 15 1 16 1 17 1 18 1 19 1 20 1 21 1 22 1 23 1 24 1 25 1 26 1 27 1 28 1 29 1 30 1 31 1 32 1 33 1 34 1 35 1 36 1 37 1 38 1 39 1 40 1 41 1 42 1 43 1 44 1 45 1 46 1 47 1 48 1 49 1 50 1

γ ,{12 }

E1

0 0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

γ ,{}

E2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

γ ,{13 }

E2

0 0 0 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44 46 47 49 50 52 53 55 56 58 59 61 62 64 65 67 68 70 71

γ ,{}

E3

0 1 2 4 6 9 12 16 20 25 30 36 42 49 56 64 72 81 90 100 110 121 132 144 156 169 182 196 210 225 240 256 272 289 306 324 342 361 380 400 420 441 462 484 506 529 552 576 600 625 650

γ ,{14 }

E3

0 0 0 0 1 2 4 7 10 14 19 24 30 37 44 52 61 70 80 91 102 114 127 140 154 169 184 200 217 234 252 271 290 310 331 352 374 397 420 444 469 494 520 547 574 602 631 660 690 721 752

γ ,{}

E4

0 1 2 4 7 11 16 23 31 41 53 67 83 102 123 147 174 204 237 274 314 358 406 458 514 575 640 710 785 865 950 1041 1137 1239 1347 1461 1581 1708 1841 1981 2128 2282 2443 2612 2788 2972 3164 3364 3572 3789 4014

γ ,{12 }

γ ,{}

E1 1
x 1 = Q0

E1 3
x 2 = Q0

γ ,{13 }

γ ,{}

E2 1
x x = Q1 x = Q0

E2

3
x 7 3x − + = Q1 2 2 3x −4 + = Q0 2 γ ,{}

E3 1
x x2 1 x + + = Q1 4 2 4 2 x x + = Q0 2 4 γ ,{14 }

E3 2
x 5x x2 2− + = Q2 3 3 5x x2 2− + = Q0 3 3 7 5x x2 − + = Q1 3 3 3 γ ,{}

25 72 1 9 1 8 2 9 17 72

+ + + + +

5x 12 5x 12 5x 12 5x 12 5x 12 5x 12

+ + + + + +

E4 1
x 5x 2 x3 + 24 36 5x 2 x3 + 24 36 5x 2 x3 + 24 36 2 5x x3 + 24 36 5x 2 x3 + 24 36 2 5x x3 + 24 36

= Q1 = Q2 = Q3 = Q4 = Q5 = Q0

650

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

Table 3 γ = {2} γ ,{}

n

E1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

γ ,{12 }

E1

0 0 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

γ ,{22 }

E1

0 0 0 0 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

γ ,{}

E2

0 0 2 3 6 7 10 11 14 15 18 19 22 23 26 27 30 31 34 35 38 39 42 43 46 47 50 51 54 55 58 59 62 63 66 67 70 71 74 75 78

γ ,{13 }

E2

0 0 0 1 3 7 10 15 18 23 26 31 34 39 42 47 50 55 58 63 66 71 74 79 82 87 90 95 98 103 106 111 114 119 122 127 130 135 138 143 146

γ ,{23 }

E2

0 0 0 0 0 0 2 4 9 12 18 21 27 30 36 39 45 48 54 57 63 66 72 75 81 84 90 93 99 102 108 111 117 120 126 129 135 138 144 147 153

γ ,{}

E3

0 0 2 4 9 14 22 30 41 52 66 80 97 114 134 154 177 200 226 252 281 310 342 374 409 444 482 520 561 602 646 690 737 784 834 884 937 990 1046 1102 1161

γ ,{}

E1 2
x 1 = Q0

γ ,{12 }

E1 4
x 3 = Q0

γ ,{22 }

E1 6
x 4 = Q0 γ ,{}

E2 2
x −2 + 2 x = Q0 −3 + 2x = Q1 γ ,{13 }

E2 5
x −13 + 4x = Q1 −14 + 4x = Q0 γ ,{23 }

E2 8
x 9x −27 + = Q0 2 57 9x − + = Q1 2 2 γ ,{}

E3 1
x 3x 2 1 −x+ 4 4 3x 2 1−x + 4 3x 2 1 −x+ 4 4 3x 2 1−x + 4 1 3x 2 −x+ 4 4 3x 2 1−x + 4

= Q1 = Q2 = Q3 = Q4 = Q5 = Q0

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

651

Table 4 γ = {1, 1} γ ,{}

n

E1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

γ ,{12 }

E1

0 0 1 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

γ ,{22 }

E1

0 0 0 0 2 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

γ ,{}

E2

0 0 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98 101 104 107 110 113 116

γ ,{13 }

E2

0 0 0 1 5 11 18 25 32 39 46 53 60 67 74 81 88 95 102 109 116 123 130 137 144 151 158 165 172 179 186 193 200 207 214 221 228 235 242 249 256

γ ,{23 }

E2

0 0 0 0 0 0 2 6 13 20 28 35 43 50 58 65 73 80 88 95 103 110 118 125 133 140 148 155 163 170 178 185 193 200 208 215 223 230 238 245 253

γ ,{}

E3

0 0 2 6 13 22 34 48 65 84 106 130 157 186 218 252 289 328 370 414 461 510 562 616 673 732 794 858 925 994 1066 1140 1217 1296 1378 1462 1549 1638 1730 1824 1921

γ ,{}

E1 2
x 1 = Q0

γ ,{12 }

E1 4
x 5 = Q0 γ ,{22 }

E1 6
x 6 = Q0 γ ,{}

E2 2
x −4 + 3x = Q0 −4 + 3x = Q1 γ ,{13 }

E2 5
x −24 + 7x = Q1 −24 + 7x = Q0 γ ,{23 }

E2 8
x 15x −47 + = Q0 2 95 15x − + = Q1 2 2 γ ,{}

E3 1
x 3 5x 2 − 2x + 4 4 5x 2 1 − 2x + 4 3 5x 2 − 2x + 4 4 5x 2 1 − 2x + 4 3 5x 2 − 2x + 4 4 5x 2 1 − 2x + 4

= Q1 = Q2 = Q3 = Q4 = Q5 = Q0

652

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

Table 5 γ = {3} γ ,{}

n E1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

γ ,{12 }

E1

0 0 0 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

γ ,{22 }

E1

0 0 0 0 1 3 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

γ ,{32 }

E1

0 0 0 0 0 0 2 4 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

γ ,{}

E2

0 0 0 2 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 103 106 109 112

γ ,{13 }

E2

0 0 0 1 2 7 13 21 28 37 44 53 60 69 76 85 92 101 108 117 124 133 140 149 156 165 172 181 188 197 204 213 220 229 236 245 252 261 268 277 284

γ ,{23 }

E2

0 0 0 0 0 0 2 5 12 20 30 39 50 59 70 79 90 99 110 119 130 139 150 159 170 179 190 199 210 219 230 239 250 259 270 279 290 299 310 319 330

γ ,{33 }

E2

0 0 0 0 0 0 0 0 0 3 6 14 22 33 42 54 63 75 84 96 105 117 126 138 147 159 168 180 189 201 210 222 231 243 252 264 273 285 294 306 315

γ ,{}

E3

0 0 0 3 6 14 25 39 56 79 102 131 163 198 236 280 324 374 427 483 542 607 672 743 817 894 974 1060 1146 1238 1333 1431 1532 1639 1746 1859 1975 2094 2216 2344 2472

γ ,{12 }

γ ,{}

E1 3
x 1 = Q0

E1 5
x 3 = Q0

γ ,{22 }

γ ,{32 }

E1 7
x 5 = Q0

E1 9
x 6 = Q0 γ ,{}

E2 4
x −8 + 3x = Q0 −8 + 3x = Q1 γ ,{13 }

E2 7
x −35 + 8x = Q1 −36 + 8x = Q0 γ ,{23 }

E2 10
x −70 + 10x = Q0 −71 + 10x = Q1 γ ,{33 }

E2 13
x 207 21x − + = Q1 2 2 21x −105 + = Q0 2 γ ,{}

E3 4
x 17x 7 x2 + = Q4 2 4 2 51 17x 7x − + = Q5 4 2 4 17x 7x 2 13 − + = Q0 2 4 2 51 17x 7x − + = Q1 4 2 4 17x 7x 2 12 − + = Q2 2 4 2 55 17x 7x − + = Q3 4 2 4

12 −

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

653

Table 6 γ = {1, 2} γ ,{}

n E1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

γ ,{12 }

E1

0 0 0 2 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

γ ,{22 }

E1

0 0 0 0 2 7 10 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

γ ,{32 }

E1

0 0 0 0 0 0 3 8 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

γ ,{}

E2

0 0 0 3 8 14 20 26 32 38 44 50 56 62 68 74 80 86 92 98 104 110 116 122 128 134 140 146 152 158 164 170 176 182 188 194 200 206 212 218 224

γ ,{13 }

E2

0 0 0 1 5 17 33 53 72 93 112 133 152 173 192 213 232 253 272 293 312 333 352 373 392 413 432 453 472 493 512 533 552 573 592 613 632 653 672 693 712

γ ,{23 }

E2

0 0 0 0 0 0 3 12 29 50 75 99 125 149 175 199 225 249 275 299 325 349 375 399 425 449 475 499 525 549 575 599 625 649 675 699 725 749 775 799 825

γ ,{33 }

E2

0 0 0 0 0 0 0 0 0 4 13 31 52 78 102 129 153 180 204 231 255 282 306 333 357 384 408 435 459 486 510 537 561 588 612 639 663 690 714 741 765

γ ,{}

E3

0 0 0 4 13 31 56 91 133 185 244 313 389 475 568 671 781 901 1028 1165 1309 1463 1624 1795 1973 2161 2356 2561 2773 2995 3224 3463 3709 3965 4228 4501 4781 5071 5368 5675 5989

γ ,{12 }

γ ,{}

E1 3
x 1 = Q0

E1 5
x 6 = Q0

γ ,{22 }

γ ,{32 }

E1 7
x 11 = Q0

E1 9
x 12 = Q0 γ ,{}

E2 4
x −16 + 6x = Q0 −16 + 6x = Q1 γ ,{13 }

E2 7
x −87 + 20x = Q1 −88 + 20x = Q0 γ ,{23 }

E2 10
x −175 + 25x = Q0 −176 + 25x = Q1 γ ,{33 }

E2 13
x 507 51x − + = Q1 2 2 51x −255 + = Q0 2 γ ,{}

E3 4
x 29 − 21x + 119 − 21x + 4 29 − 21x + 119 − 21x + 4 29 − 21x + 119 − 21x + 4

17x 2 4 17x 2 4 17x 2 4 17x 2 4 17x 2 4 17x 2 4

= Q4 = Q5 = Q0 = Q1 = Q2 = Q3

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Table 7 γ = {13 } γ ,{}

n E1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

γ ,{12 }

E1

0 0 0 4 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 γ ,{}

E3 4
x

γ ,{22 }

E1

0 0 0 0 4 13 18 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19

γ ,{32 }

E1

0 0 0 0 0 0 5 14 19 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

γ ,{}

γ ,{13 }

γ ,{23 }

γ ,{33 }

E2

E2

E2

E2

0 0 0 5 14 24 34 44 54 64 74 84 94 104 114 124 134 144 154 164 174 184 194 204 214 224 234 244 254 264 274 284 294 304 314 324 334 344 354 364 374

0 0 0 1 9 31 63 99 136 173 210 247 284 321 358 395 432 469 506 543 580 617 654 691 728 765 802 839 876 913 950 987 1024 1061 1098 1135 1172 1209 1246 1283 1320

0 0 0 0 0 0 5 22 53 94 139 185 231 277 323 369 415 461 507 553 599 645 691 737 783 829 875 921 967 1013 1059 1105 1151 1197 1243 1289 1335 1381 1427 1473 1519

0 0 0 0 0 0 0 0 0 6 23 55 96 142 188 235 281 328 374 421 467 514 560 607 653 700 746 793 839 886 932 979 1025 1072 1118 1165 1211 1258 1304 1351 1397

γ ,{}

E3

0 0 0 6 23 55 102 165 243 337 446 571 711 867 1038 1225 1427 1645 1878 2127 2391 2671 2966 3277 3603 3945 4302 4675 5063 5467 5886 6321 6771 7237 7718 8215 8727 9255 9798 10357 10931

γ ,{12 }

γ ,{}

E1 3
x 1 = Q0

E1 5
x 10 = Q0

γ ,{22 }

γ ,{32 }

E1 7
x 19 = Q0

E1 9
x 20 = Q0 γ ,{}

E2 4
x −26 + 10x = Q0 −26 + 10x = Q1 γ ,{13 }

E2 7
x −160 + 37x = Q1 −160 + 37x = Q0 γ ,{23 }

E2 10
x −321 + 46x = Q0 −321 + 46x = Q1 γ ,{33 }

E2 13
x 93x − 925 2 + 2 = Q1 −463 + 93x 2 = Q0

2 51 − 38x + 31x 4 = Q4

2 51 − 38x + 31x 4 = Q0

2 51 − 38x + 31x 4 = Q2

205 − 38x + 31x 2 = Q 5 4 4

205 − 38x + 31x 2 = Q 1 4 4

205 − 38x + 31x 2 = Q 3 4 4

J.O. Carbonara et al. / Journal of Algebra 260 (2003) 631–656

655

Table 8   Crδ (x) = sλ ⊗ sλ , s(δ,x−|δ|) λ x l(λ)
r

 −48 − 4x + 5x 2 + x 3    −5 − x + 5x 2 + x 3      −20 − 4x + 5x 2 + x 3     −21 − x + 5x 2 + x 3      −32 − 4x + 5x 2 + x 3   2 3 (48) C41 (x) = −5 − x + 5x +2 x 3  + x −36 − 4x + 5x    2 3   −5 − x + 5x + x    −32 − 4x + 5x 2 + x 3      −21 − x + 5x 2 + x 3   2 3    −20 − 4x + 5x + x 2 3 −5 − x + 5x + x

if x ≡ 0 mod (12) if x ≡ 1 mod (12) if x ≡ 2 mod (12) if x ≡ 3 mod (12) if x ≡ 4 mod (12) if x ≡ 5 mod (12) if x ≡ 6 mod (12) if x ≡ 7 mod (12) if x ≡ 8 mod (12) if x ≡ 9 mod (12) if x ≡ 10 mod (12) if x ≡ 11 mod (12)

 −6 + x 2 if x ≡ 0 mod (6)      −1 + x 2 if x ≡ 1 mod (6)   2 (6) C31 (x) = −4 + x 2 if x ≡ 2 mod (6)  if x ≡ 3 mod (6) −3 + x    2 if x ≡ 4 mod (6)   −4 + x  −1 + x 2 if x ≡ 5 mod (6)  −2 + x if x ≡ 0 mod (2) (2) C21 (x) = −1 + x if x ≡ 1 mod (2)

The functions above are valid for x  1   −6 + x if x ≡ 0 mod (2) 24 − 15x + 2x 2 if x ≡ 0 mod (6)   C23 (x) =  2  −5 + x if x ≡ 1 mod (2)  if x ≡ 1 mod (6) 25 − 15x + 2x   2 if x ≡ 2 mod (6) 22 − 15x + 2x 3 (2) C3 (x) =  27 − 15x + 2x 2 if x ≡ 3 mod (6)      22 − 15x + 2x 2 if x ≡ 4 mod (6)  25 − 15x + 2x 2 if x ≡ 5 mod (6)   −4 + x if x ≡ 0 mod (2) 30 − 20x + 3x 2 if x ≡ 0 mod (6)  1,2  C (x) =   2 −4 + x if x ≡ 1 mod (2)  33 − 20x + 3x 2 if x ≡ 1 mod (6)   2 if x ≡ 2 mod (6) 1,2 32 − 20x + 3x (2) C3 (x) = 2 if x ≡ 3 mod (6)   31 − 20x + 3x   2    32 − 20x + 3x 2 if x ≡ 4 mod (6) if x ≡ 5 mod (6) 33 − 20x + 3x   2 −2 + x if x ≡ 0 mod (2) 30 − 21x + 4x if x ≡ 0 mod (6)  1,1,1  (2) C (x) =   2 −3 + x if x ≡ 1 mod (2)  23 − 21x + 4x 2 if x ≡ 1 mod (6)   2 if x ≡ 2 mod (6) 1,1,1 26 − 21x + 4x (6) C3 (x) =  27 − 21x + 4x 2 if x ≡ 3 mod (6)    2    26 − 21x + 4x 2 if x ≡ 4 mod (6) if x ≡ 5 mod (6) 23 − 21x + 4x  2 − 3x + x 2      −3x + x 2   2 (2) C32 (x) = 2 − 3x +2x  −3x + x    2   2 − 3x + x  −3x + x 2  3 − 3x + x 2      2 − 3x + x 2   2 (3) C31,1 (x) = 2 − 3x + x 2  3 − 3x + x    2    2 − 3x + x 2 2 − 3x + x

The functions above are valid for x  4  −2 + x if x ≡ 0 mod (2) if x ≡ 0 mod (6) C22 (x) = −3 + x if x ≡ 1 mod (2) if x ≡ 1 mod (6) if x ≡ 2 mod (6) if x ≡ 3 mod (6) if x ≡ 4 mod (6) if x ≡ 5 mod (6)  −2 + x if x ≡ 0 mod (2) if x ≡ 0 mod (6) (2) C21,1 (x) = −1 + x if x ≡ 1 mod (2) if x ≡ 1 mod (6) if x ≡ 2 mod (6) if x ≡ 3 mod (6) if x ≡ 4 mod (6) if x ≡ 5 mod (6) The functions above are valid for x  2

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