COMBINATORICS OF REGULAR PARTITIONS

COMBINATORICS OF REGULAR PARTITIONS

arXiv:1408.4866v1 [math.CO] 21 Aug 2014

HIROSHI MIZUKAWA AND HIRO-FUMI YAMADA

Abstract. Two partition identities are given, which are concerned with r-regular partitions and r-class regular partitions. Together with some discussions on the HallLittlewood and the Kostka-Foulkes polynomials, these formulas lead to the computation of the r-regular character table of the symmetric group.

1. Introduction Let r be an integer greater than 1. A partition is said to be r-regular if no part is repeated r or more times, and is said to be r-class regular if no part is divisible by r. In this short note we present an “r-congruence property” of r-regular / r-class regular partitions (Theorem 2.1), and an “r-adic property” of r-class regular partitions (Theorem 3.1). We will prove these formulas by manipulation of generating functions. Next we consider the ordinary character table if the symmetric group Sn . More precisely we compute the determinant of the “regular character table”, which consists of the values of irreducible characters corresponding to the r-regular partitions on r-regular conjugacy classes. Determinant of the regular character table has been computed by Olsson [6] (see also [1]). We will give another proof of Olsson’s result, by examining the transition matrices of the Hall-Littlewood symmetric functions and the Schur functions. When r is prime, it is known that the determinant of the regular character table equals that of the r-Brauer character table. 2. regular partitions Throughout this note, we fix a positive integer r ≥ 2. For n ≥ 0, let Pn be the set of partitions of n. We denote by mi (λ) the multiplicity of i in λ ∈ Pn . A partition λ is said to be an r-regular if mi (λ) < r holds for any i ≥ 1. A partition ρ is said to be an r-class regular if all parts are not divisible by r. Let RPr,n (resp. CPr,n ) be the set of r-regular (resp. r-class regular) partitions. A classical result says that RPr,n and CPr,n have the 2010 Mathematics Subject Classification. Primary: 05E10; Secondary:05E05 . Key words and phrases. r-regular partition, r-class regular partition, Hall-Littlewood symmetric function, character table of symmetric group. The first author was supported by KAKENHI 24740033. The second author was supported by KAKENHI 24540020. 1

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MIZUKAWA AND YAMADA

same cardinality. The generating function X X RPr,n q n = CPr,n q n Φr (q) = n≥0

n≥0

has a product expression

Φr (q) =

Y 1 − q rk

k≥1

1 − qk

.

For a partition λ = (λ1 , λ2 , . . .) and j ∈ {1, 2, . . . , r − 1}, we put xr,j (λ) = {i | λi ≡ j (mod r)} and yr,j (λ) = {i | mi (λ) ≥ j} .

We define

Xr,j,n =

X

xr,j (ρ) and Yr,j,n =

X

yr,j (λ).

λ∈RPr,n

ρ∈CPr,n

Theorem 2.1. The number Xr,j,n − Yr,j,n is a non-negative integer independent for j = 1, 2, . . . , r − 1, and equal to cr,n , the coefficient of q n in Φr (q)

X k≥1

q rk . 1 − q rk

Before proving this theorem, we give an example. Example 2.2. We take r = 3 and n = 7. The following table lists the 3-class regular partitions of 7: ρ x3,1 (ρ) x3,2 (ρ)

7 52 512 421 413 23 1 22 13 215 17 total 1 0 2 2 4 1 3 5 7 25 0 2 1 1 0 3 2 1 0 10

From the table above, we have X3,1,7 = 25 and X3,2,7 = 10. As for the 3-regular partitions of 7, we have λ i≥1 mi (λ)! y3,1 (λ) y3,2 (λ)

Q

7 61 52 512 52 421 32 1 322 3212 total 1 1·1 1·1 1·21 1·1 1·1·1 21·1 1·21 1·1·21 − 1 2 2 2 2 3 2 2 3 19 0 0 0 1 0 0 1 1 1 4

From the second table, we have Y3,1,7 = 19 and Y3,2,7 = 4. Thus we see X3,1,7 − Y3,1,7 = X3,2,7 − Y3,2,7 = 6. On the other hand we have X q 3k Φ3 (q) = q 3 + q 4 + 2q 5 + 4q 6 + 6q 7 + 9q 8 + 13q 9 + 19q 10 + · · · . 3k 1 − q k≥1


3

Proof. First we will compute the generating function of Xr,j,n . For i 6≡ 0 (mod r), we have X P 1 − qi ρ∈CPr,n mi (ρ) Φr (q) qn. = t 1 − tq i n≥0 Taking the t-derivative at t = 1, we obtain i

Φr (q)

Since xr,j (ρ) =

P

k≥0 mkr+j (ρ),

q = 1 − qi

X n≥0

 

X

ρ∈CPr,n



mi (ρ) q n .

(2.1)

we have the following generating function Xr,j (q) of Xr,j,n.

Xr,j (q) =

X

Xr,j,n q n = Φr (q)

n≥0

X k≥0

q rk+j . 1 − q rk+j

Second, we consider the r-regular partitions and the generating function of Yr,j,n. We put Φr,j (q, t) =

Y

(1 + q k + q 2k + · · · + q (j−1)k + tq jk + tq (j+1)k + · · · + tq (r−1)k ).

k≥1

Immediately we have

Φr,j (q, t) =

X n≥0

 

X

λ∈RPr,n



tyr,j (λ)  q n .

Taking the t-derivative at t = 1, we obtain   X X X d  yr,j (λ) q n = Φr,j (q, t) = Yr,j,n q n . dt t=1 n≥0 n≥0 λ∈RP r,n

As for the equation (2.2), we have

X q jk − q rk d Φr,j (q, t) = Φr,j (q, t) . dt 1 − q rk t=1 k≥1

The generating function Yr,j (q) of Yr,j,n reads Yr,j (q) =

X n≥0

Yr,j,nq n = Φr (q)

X q jk − q rk k≥1

1 − q rk

.

(2.2)

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MIZUKAWA AND YAMADA

To complete the proof, we compute X

Xr,j (q) − Yr,j (q) = Φr (q)

(q rk+j + q 2(rk+j) + q 3(rk+j) + · · · )

k≥0

−

X

(q jm + q (r+j)m + q (2r+j)m + · · · ) +

m≥1

X k≥1

XX

= Φr (q)

q m(rk+j) −

k≥0 m≥1

= Φr (q)

X k≥1

XX

q rk 1 − q rk

q (kr+j)m +

m≥1 k≥0

X k≥1

X q rk = cr,n q n . 1 − q rk n≥0

!

q rk 1 − q rk

!

The concluding q-series is independent of j. Therefore Xr,j,n − Yr,j,n does not depend on the choice of j. 3. Class regular partitions We define, for j ≥ 1, Vr,j,n =

X

mj (ρ) and Wr,j,n =

X

yr,j (ρ).

ρ∈CPr,n

ρ∈CPr,n

Note that the sum is taken over r-class regular partitions of n both in Vr,j,n and Wr,j,n. Theorem 3.1. For j 6≡ 0 (mod r), we have X Wr,ri j,n . Vr,j,n = i≥0

Before proving this theorem, we give an example. Example 3.2. There are sixteen 4-class regular partitions of n = 8. The following table lists Vr,j,n and Wr,j,n of them: j V4,j,8 W4,j,8

1 2 3 4 5 6 7 8 38 16 8 0 3 2 1 0 33 15 8 5 3 2 1 1

We have V4,1,8 = W4,1,8 + W4,4,8 and V4,2,8 = W4,2,8 + W4,8,8 . Proof. From (2.1), we have the generating function of Vr,j,n: X n≥0

Vr,j,nq n = Φr (q)

qj . 1 − qj


5

Let k ≡ 6 0 (mod r). We have X {ρ ∈ CPr,n | mk (ρ) ≥ j} q n = Φr (q)(1 − q k )(q jk + q (j+1)k + q (j+2)k + · · · ) n≥0

= Φr (q)q jk .

Thus we have the generating function of Wr,j,n: X X Wr,j,nq n = Φr (q) n≥0

q jk

k6≡0 (mod r)

= Φr (q)

X (q jk − q rjk ) k≥1

qj q rj = Φr (q) − 1 − qj 1 − q rj

.

Hence the generating function of RHS of the claim reads ! !! ri j r i+1 j X X X X q q Φr (q) Wr,rij,n q n = − 1 − q ri j 1 − q ri+1 j n≥0 i≥0 n≥0 i≥0 = Φr (q)

qj 1 − qj

(3.1)

as desired.

Theorem 3.1 implies the following formulas. X X Corollary 3.3. (1) cr,n = iWr,ri j,n . (2)

Y Y

mi (ρ)! =

ρ∈CPr,n i≥1

j6≡0 (mod r) i≥1 Y Y cr,n ρi . r ρ∈CPr,n i≥1

Proof. Since Vr,j,n = 0 for any j 6≡ 0 (mod r), we have Y Y Y Y ρi = j Vr,j,n = ρ∈CPr,n i≥1

j≥1

On the other hand we compute Y Y Y mi (ρ)! = j Wr,j,n = ρ∈CPr,n i≥1

j≥1



=

Y

j6≡0 (mod r)

Y

= r dr,n

Y (r i j)Wr,rij,n

j6≡0 (mod r) i≥0

Y

j6≡0 (mod r) i≥0

Y

j Vr,j,n .





r iWr,rij,n  × 

j

P

i≥0

Wr,r i j,n

j6≡0 (mod r)

Y = r dr,n j Vr,j,n , Thm.3.1 j6≡0 (mod r)

Y

Y

j6≡0 (mod r) i≥0



j Wr,ri j,n 

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MIZUKAWA AND YAMADA

where we have put X

dr,n =

X

iWr,ri j,n .

j6≡0 (mod r) i≥1

We verify that dr,n = cr,n as follows. ! ! X X X X i Wr,ri j,n q n iWr,ri j,n q n = n≥0

i≥0

i≥0

n≥0

i

=

(3.1)

X

iΦr (q)

i≥0

= Φr (q)

i+1

qr j qr j − 1 − q ri j 1 − q ri+1 j

!

i

X i≥1

qr j . 1 − q ri j

Now we take sum over j 6≡ 0 (mod r) to have the generating function of dr,n . ! X X X q ri j dr,n q n = Φr (q) 1 − q ri j n≥0 i≥1 j6≡0 (mod r)

= Φr (q)

X n≥1

=

X

q rn 1 − q rn

cr,n q n .

n≥0

The formula (2) in Corollary 3.3 is due to Olsson [6]. In the case of prime r, arithmetic of r-regular / r-class regular partitions is closely studied by Bessenrodt and Olsson [1] in connection with the modular representations of the symmetric groups. 4. Regular partitions and Hall-Littlewood symmetric functions In this section, we apply Theorem 2.1 to computations of some minor determinants of transition matrices and the character tables of the symmetric groups. 4.1. Hall-Littlewood symmetric functions at root of unity. The Hall-Littlewood P - and Q- symmetric functions ([4]) are a one parameter family of symmetric functions satisfying the orthogonality relation: hPλ (x; t), Qµ (x; t)it = δλµ , where the inner product h, it is defined by hpλ (x), pµ (x)it = zλ (t)δλµ with zλ (t) = zλ tλi )−1 . Let (a; t)n be a t-shifted factorial:  (1 − a)(1 − at) · · · (1 − atn−1 ) (n ≥ 1) (a; t)n = 1 (n = 0).

Q

i≥1 (1−


7

The relation between P - and Q- functions is described as Qλ (x) = bλ (t)Pλ (x), where bλ (t) =

Q

i≥1 (t; t)mi (λ) .

4.2. Q′ -functions. In this note, we are interested in the case that parameter t is a primitive r-th root of unity ζ. The Hall-Littlewood symmetric functions at root of unity is studied at the first time by [5]. We remark that {Qλ (x; ζ) | λ ∈ RPr,n } is a Q(ζ)-basis for the subspace Λ(r) = Q(ζ)[ps (x) | s 6≡ 0 (mod r)] of the symmetric function ring Λ = Q(ζ)[ps (x) | s = 1, 2, . . .]. This can be shown along the arguments in [4, Chap. 3-8], where the case r = 2 is discussed. In [3], Lascoux, Leclerc and Thibon consider the dual basis (Q′λ ) of P -functions, relative to the inner product at t = 0. Namely P - and Q′ functions satisfy the Cauchy identity: X Y (1 − xi yj )−1 . Pλ (x; t)Q′λ (y; t) = i,j

λ

When t = ζ, the Q′ -functions have the following nice factorization property. Proposition 4.1 ([3]). Let ζ be a primitive r-th root of unity. If a partition λ satisfies mi (λ) ≥ r, then we have Q′λ (x; ζ) = (−1)i(r−1) Q′λ\(ir ) (x; ζ)hi (xr ). Here hi (xr ) = hi (xr1 , xr2 , . . .) and λ \ (ir ) is a partition obtained by removing the rectangle (r i ) from the Young diagram λ. We define an r-reduction for a symmetric function f (x) by f (r) (x) = f (x) pr (x)=p2r (x)=p3r (x)=...=0 .

Proposition 4.1 leads us to the following lemma. (r)

Lemma 4.2. Q′ λ (x; ζ) = 0 unless λ is an r-regular partition. We set Qλ (x; ζ) =

X

(r)

Qλρ pρ (x) and Q′ λ (x; ζ) =

X

ρ∈CPr,n

ρ∈CPr,n

Proposition 4.3. Let λ ∈ RPr,n and ρ ∈ CPr,n . We have Y λ Q′ ρ = (1 − ζ ρi )−1 Qλρ . i≥1

λ

Q′ ρ pρ (x).

8

MIZUKAWA AND YAMADA

Proof. We compute inner products at t = ζ and t = 0 for r-regular partitions λ and µ. Namely, we see X Qλρ Qµρ zρ (ζ) δλµ = hPλ (x; ζ), Qµ(x; ζ)iζ = bλ (ζ)−1 ρ∈CPr,n

and (r)

δλµ = hPλ (x; ζ), Q′µ (x; ζ)i0 = bλ (ζ)−1

X

µ

Qλρ Q′ ρ zρ .

ρ∈CPr,n

Since {Pλ (x; ζ) | λ ∈ RPr,n } is also a basis of Λ(r) , we have the claim.

We define Lλµ (t) by sλ (x) =

X

Lλµ (t)Q′µ (x; t),

µ∈Pn

where sλ (x) denotes the Schur function. Let Kλµ (t) be the Kostka-Foulkes polynomial ([4]). In other words, the matrix K(t) = (Kλµ (t))λ,µ∈Pn is the transition matrix M(s, P ) from the Schur functions to the Hall-Littelewood P -functions. It is known that K(t) is an upper unitriangular matrix. (−1)

Lemma 4.4. For partitions λ and µ, we have Lλµ (t) = Kµλ (t), the (λ, µ)-entry of the matrix K(t)−1 . Proof. Lλµ (t) = hsλ (x), Pµ (x; t)i0 = hsλ (x),

X

(−1)

(−1) Kµν sν i0 = Kµλ (t).

ν∈Pn

Example 4.5. We take ζ = −1 and n = 4. Then we have s4 (x) = Q′ 4 (x; −1), s31 (x) = Q′ 31 (x; −1) + Q′ 4 (x; −1), s22 (x) = Q′ 22 (x; −1) + Q′ 31 (x; −1), s211 (x) = Q′ 211 (x; −1) + Q′ 22 (x; −1) + Q′ 31 (x; −1) + Q′ 4 (x; −1), s1111 (x) = Q′ 1111 (x; −1) + Q′ 211 (x; −1) − Q′ 22 (x; −1) + Q′ 4 (x; −1). Lemma 4.2 and 4.4 give the following expansion formula. Proposition 4.6. Let λ ∈ Pn and µ ∈ RPr,n . We have X (r) (−1) (r) Kµλ (ζ)Q′µ (x; ζ). sλ (x) = µ∈RPr,n

In particular, (Lλµ (ζ))λ,µ∈RPr,n is a lower unitriangular matrix.


9

Example 4.7. By Proposition 4.6, we immediately see (2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

s4 (x) = Q′ 4 (x; −1), (2)

s31 (x) = Q′ 31 (x; −1) + Q′ 4 (x; −1), s22 (x) = Q′ 31 (x; −1), (2)

s211 (x) = Q′ 31 (x; −1) + Q′ 4 (x; −1), (2)

(2)

s1111 (x) = Q′ 4 (x; −1). From the first two equations above, we have !

1 0 . 1 1

(Lλµ (−1))λ,µ∈RP2,4 =

We set (r)

s(r) = {sλ (x) | λ ∈ RPr,n }, Q′

(r)

(r)

= {Q′ λ (x) | λ ∈ RPr,n }

and p(r) = {pλ (x) | λ ∈ CPr,n }. For u, v ∈ {s(r) , Q′ (r) , p(r) }, we denote by M(u, v) the transition matrix from u to v. By Lemma 4.2, we have that M(s(r) , Q′ (r) ) is obtained by removing non r-regular rows and columns from the transposed inverse of K(t). Theorem 4.8. We have det M(Q′(r) , p(r) ) ∈ R and

det M(Q′(r) , p(r) ) = ±

1 r cr,n

Q

ρ∈CPr (n)

Q

i≥1

ρi

.

Proof. The orthogonality relation of Pλ (x; ζ) and Q′µ (x; ζ): (r)

δλµ = hPλ (x; ζ), Q′µ (x; ζ)i0 = bλ (ζ)−1

X

ρ∈CPr,n

µ

Qλρ Q′ ρ zρ ,

10

MIZUKAWA AND YAMADA

and Proposition 4.3 give 2 Q 1  λ∈RPr,n bλ (ζ) Q det M(Q′(r) , p(r) )2 =  1 − ζ ρi ρ∈CPr,n zρ (ζ) ρ∈CPr,n i≥1 Q bλ (ζ) λ∈RP Q r,n Q =Q ρi ρ∈CPr,n zρ ( ρ∈CPr,n i≥1 (1 − ζ )) Q Q mi (λ) ) λ∈RPr,n i≥1 (1 − ζ Q Q =Q ρi ρ∈CPr,n zρ ( ρ∈CPr,n i≥1 (1 − ζ )) Q Qr−1 j yr,j (λ) λ∈RPr,n j=1 (1 − ζ ) =Q Q Qr−1 j xr,j (ρ) ) ρ∈CPr,n zρ ( ρ∈CPr,n j=1 (1 − ζ ) Qr−1 j Yr,j,n j=1 (1 − ζ ) =Q Qr−1 j Xr,j,n j=1 (1 − ζ ) ρ∈CPr,n zρ 

Y Y

=Q

ρ∈CPr,n

=Q

ρ∈CPr,n

zρ zρ

1 j Xr,j,n −Yr,j,n j=1 (1 − ζ )

Qr−1 1 Qr−1 j=1

(1 − ζ j )cr,n

We apply Theorem 2.1 to the last equality above. By noticing Corollary 3.3 (2), we obtain the formula.

.

Qr−1

i=1 (1 − ζ

i

) = r and using

4.3. Regular character tables of the symmetric groups. Let Tn = (χλρ )λ,ρ∈Pn be the ordinary character table of the symmetric group Sn . The orthogonality relation of the characters implies Y zρ . (det Tn )2 = ρ∈Pn

From James’s book [2, Corollary 6.5], this formula can be simplified as (det Tn )2 =

YY

ρ2i .

ρ∈Pn i≥1 (r)

Olsson considers the r-regular character table Tn = (χλρ )λ∈RPr,n ,ρ∈CPr,n and computes its determinant. He proves the following theorem. Theorem 4.9 ([6]). det Tn(r) = ±

Y Y

ρ∈CPr,n i≥1

ρi .


11

Proof. Theorem 4.8 and Proposition 4.6 enable us to compute the determinant of the regular character table as follows: (r)

(r)

det M(s(r) , p(r) )2 = det M(s(r) , Q′ )2 det M(Q′ , p(r) )2 =1×

1 r 2cr,n

Since det M(s(r) , p(r) )2 = (det Tn(r) )2 ×

Y

zρ−2

ρ∈CPr,n

we have



(det Tn(r) )2 = 

Q

ρ∈CPr,n

=

Cor.3.3(2)

Q

i≥1

ρ2i

(det Tn(r) )2 × r −2cr,n

Y Y

ρ∈CPr,n i≥1

Y Y

ρ−4 i ,

ρ∈CPr,n i≥1

2

ρi  .

References [1] C. Bessenrodt and J. B. Olsson, Submatrices of character tables and basic sets, J. Combinatorial Th. Ser. A no. 119 (2012), 1774-1788. [2] G. James, The representation theory of the symmetric groups, Lecture notes in mathematics 682, Springer-Verlag 1978. [3] A. Lascoux, B. Leclerc and J.-Y. Thibon Fonctions de Hall-Littlewood et polynômes de Kostka-Foulkes aux racines de l’unité., C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no.1, 1-6. [4] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd. ed. , Oxford, Clarendon Press, 1995. [5] A. O. Morris, On an algebra of symmetric functions, Quart. J. Math. Oxford Ser. (2) no.16 (1965) 53-64. [6] J. B. Olsson, Regular character tables of symmetric groups, Electron. J. Combin. 10 (2003) N3. MR1975776. Hiroshi Mizukawa, Department of Mathematics, National Defense Academy of Japan, Yokosuka 239-8686, Japan E-mail address: [email protected] Hiro-Fumi Yamada, Department of Mathematics, Okayama University, Okayama 7008530, Japan E-mail address: [email protected]