奈良教育大学学術リポジトリNEAR
Note on the Symmetric Functions
著者 OCHIAI Shoji
journal or
publication title
奈良教育大学紀要. 自然科学
volume 27
number 2
page range 1‑3
year 1978‑11‑25
URL http://hdl.handle.net/10105/2482
奈良教育大学紀要 第27巻 第2号(自然)昭和53年 Bull. Nara Univ. Educ, Vol. 27, No. 2 (Nat.), 1978
Note on the Symmetric Functions
Shoji OCHIAI
Department of Mathematics, Nara University of Education, Nara, Japan (Received May 1, 1978)
1. Introduction. Let ♂1,♂2, ‑ ,♂〃 be the elementary symmetric functions in the inde‑
terminates X¥,x2,〜,xn. It is classical that every symmetric function of x¥,x2,・・・,xn over
integers (or丘eld) is a polynomial in the elementary symmetric functions ♂1, ♂2, ･‑, ♂打.
In this note, we will give the computational technique which expresses a given sym‑
metric function by a polynomial in the elementary symmetric functions ♂1,♂2,〜,♂. Our interest in this problem originates from the computation of the Steenrod reduced powers in the cohomology of classifying space for the classical Lie groups. The related results are found in 〔2〕, 〔3〕.
2. Statements of results and the proofs. Let A be amatrixof m rowsand n colums with elements 0 or 1. This matrix A‑(chj) is called (0, 1) matrix of size m by n. We
call R‑(ri,r2,・・・,rm), S‑(si,s2,・蝣蝣, sn) the row sum vector, the column sum vector of A where
II m
r;‑∑ai.j, Sj‑∑aij. We denote the set of all (0, 1) matrix of size m by n with the
J‑I 1‑1
prescribed row sum vector R and the prescribed column sum vector S by a{R;S) and the
number of elements of α(R;S) by (R;S).
h
Let (o be a partition of N,i.e.a)‑(ni,n2,〜,m) ∑m‑N,?ii≧ ‑ ≧nk≧1. Then we denote
l=
the conjugate partition of co by cサ*. The next Lemma is trivial.
Lemma 1. (co* ;<o)‑l holds for any partition co.
It seems diffucult to determine the precise number (R;S) for general R and S. The next reduction formula is su伍cient for our purpose.
Lemma 2. Let R‑(ri,r2,‑‑‑rm), S‑(sus2, ‑ 5B) be the row sum vector, column sunてector
of the (0, 1) matrix A, respectively. Then ‑we have the next reduction formula.
{(ri,r2, ‑,rm) ; (si,s2, ‑,sn))‑∑ ((^2,r3, ‑ rm) ; [Si‑di,s2‑∂2,〜,Sn‑dn))
n
where the sum is extended over all (♂:,82,・蝣蝣,dn) such that. ∑∂i‑T¥, Si‑♂,･≧0, ♂,‑0 or 1.
1‑1
We understand ((n,r,+i,・・・,rm) ; (Si′,s2',・・・,sn'))‑0 if the number of thepositive integers of
the set {si′, s2',・蝣, 5サ'}<ry for somej {i≦j≦m) and ((r);(si', s2',〜, 5B')) ‑1 ifsi'‑0 or lfor any i.
れ
Proof. Any matrix of a(R;S) withthe first row ∂.,d2,・・・,8n) must satisfy ∑∂ ‑J‑i, and
l.‑‖
Si‑♂,.≧0, ♂,‑1 Or o, for any i. The number of such (0, 1) matrices is obviously (r2,r3, Si‑∂1,$2‑ ∂2, 〜,sn‑ ∂ This implies the theorem.
Let 22‑{co} be the set of all partitions of a positive integer N, where o)‑(ki,k2,
Iれ
･,kn ∑h‑N, h≧k2≧ ･･･≧km≧1. Let co‑(ki,k2,〜,kn), v‑{luh, ‑ lm)bethe partitions
H:+り
of 盟. De丘ne the order in 盟 as follows.
1
2 Shsii Oohiai
(o<C,vifki<Chorki‑li,k2,‑l2,‑‑‑,ki‑i‑lLki<li.
Letco‑(ki,ki,蝣・蝣,kn)beanelementofQand∑x¥xx¥*・・・・｣"bethesmallestsymmetric functioncontainingtheterm;君1X2‑3￡Thenbytheclassicaltheorem,wecanexpress
∑Ef.砕‑∬たn‑∑a"帝,l)帝,2)‑̀蔚*coD c｡)‑･‑(B)
wherethesumisextendedoverallmonomialsofweightNanddimensionkiandplCO,
A(i,1)arethepositiveintegerswithju{1)≧M(2)≧‑≧/*(*(*))≧1.Thereforeanunsettled partistodeterminecoe伍cientsen.Fromtherighthandside,weconstructthesetof
partitionofN, W,l)W,2)k{i,kit)) ー′‑｣ヽヽ′⊥,
Q'‑W),‑,/ォ(1),ju(2),‑,〟(2),〜,M(*(i)),‑,/m{k(i))}.ThesubsetSi'ofQisalreadyordered bytheabovementionedrule.
Hi,1)W,2)W,(W) 巳===己==コ
Weputβ(i)‑(M(l),‑,M(D,M(2),‑,rt2),‑,fi{k(i)),‑*(*(*))),1≦i≦q(qisthenumberof partitionsofki+Nintothekiparts).Ifβ(*蝣)*‑W?5,^芸')...鵜〕),耳)≧*2≧･.･≧鵜)≧1,
‑eoperate品)私･･(品V2I品.3(0on(B),i‑I,2,‑q.Then‑egetthesyste‑
ofqlinearequations l‑(β(1)"β(1)α1
0‑(β(2)*;β(D)α1+(β(2)ォβ(2))α2
0‑(β(gY β(!))<*!+ ‑ +(β(gV β{q))a, Therefore we get the next Theorem by Cramer's rule.
Theorem. The coefficient a/ 1≦f≦q is given by (ou)
(0(2):霊ト＼､0:
"iZI､:
0
‑
‑
‑ 0
(β(∫‑1)*; β(∫‑1))
1
1
I
l
(β(?)*; β(D) ‑ (β(qY β(/‑D)
0
(β(∫+1)*; β(∫+1))
ヽヽ 己■
6コ ヽヽ
､ヽ
03(9)*; β(/+!)) ‑ (β(?)*; β(<?))
3. Applications. We will apply the Theorem to compute the Steenrod reduced power operations in the cohomology of the classifying space of Lie groups. We will treate one
case and others are treated similary.
Corollary. If C2>‑ ∈ H2i{BU{n) ; Z3) denote the Chern classes mod 3, we obtain 91C* ‑ C‑?Ck ‑ 2C42‑ C2Ce+ C8
Proof. If we express the symmetric function ∑ x¥ x2 by a polynomial of the elementary symmetric functions, the proof is completed by Borel and Serre's results 〔1〕 We can see easily β(1)‑(2,1, 1), β(2)‑(2,2), β(3)‑(3, 1), β(4)‑(4). In this case, the matrix of Theorem is
Note on the Symmetric Functions
((3,1) ; (2,1,1))
((2,2) ; (2,1,1)) ((2,2) ; (2,2,1))
((2,1,1) ; (2,1,1)) ((2,1,1) ; (2,2)) ((2,1,1) ; (3,1))
((1,1,1,1); (2,1,D) ((1,1,1,1) ; (2,2)) ((1,1,1,1); (3,1)) ((1,1,1,1); (4))
3
Using Lemma 2, we get ((2,2) ; (2,1,1))‑((2) ; (2,0,0))+((2) ; (1,0,l))+((2) ; (0,1,1))‑2, ((2,1,1)) ; (2,1,1))‑5, ((2,1,1) ; (2,2))‑2, ((1,1,1,1) ; (2,1,1))‑12, ((1,1,1,1) ; (2,2))‑6, ((1,1,1,1)) ; (3,1))‑4. According to Theorem, we obtain ai‑l, a2‑‑2, a3‑‑1, a4‑4.
Therefore, we can see
∑ x^xi‑tf2(?i2‑2a2 ‑a3ffi+4at This concludes the proof.
References
〔1〕 A. Borel and J. P. Serre, Gronpes de hie et puissances rcduites de Steenrod, Amer. J. Math.
75 (1953) 409‑448.
〔2〕 S. Jimi and T. Sugawara, Reduced power operations on H*(B均n),Zp), (p‑3), Memories of the Faculty of Science, Kyushu University, Ser. A, 26, no. 2, (1972) 285‑291.
〔3〕 S. Mukoda and S. Sawaki, On the bpk''coefficient ofa certain symmetric functions, J. Fac. Sci.
Niigata Univ. 1. (1954)ト6.
