Osima, Masaru

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On some character relations of symmetric groups

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 1, Issue 1 1952 Article 3MARCH 1952On some character relations of symmetricgroupsMasaru Osima∗∗Copyright c©1952 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouON SOME CHARACTER RELATIONS OFSYMMETRIC GROUPSl\lAsARu OSIMA1. Let 11, be a natural number and let(1 ) n = a 1 + £(2 + ' + all. ,be a partition (a i ) of n into h natural numbers al. By a diagramT = [a l ] con'esponding to this partition we mean an arrange-ment of n nodes into h rows consisting of a!' £(~, ••• "', aT!. nodes.The number men) of distinct diagrams of 1l nodes is equal to thenumber of irreducible representations of the symmetric group ®n.We set tn(O) = 1. If P is a fixed prime number, then the numbers(n) of diagrams without P-hook1) is equal to the number of P-blocksof highest kind:!). Let( 2 ) n = kp + r, o s;:: r < p.kBy R. Brauer:n, the number of P-blocks of @Sa is equal to ~ s(n-J.p).A=(INow we define 1(A.) and 1*(J.) by(3 )(4 )leA)l*(A) ~ tn(1I 1)1n(v2) ••• "'1n(v p _J'Ill. \12 •.•..••• \lp-t(::E Ai = J., 0 < A. i < A)(~ lJi = J., 0 s;:: lJ i ~ J.).Let To be a diagram of ell-AV without P-hook. Then To determinespniquely a P-block B rr of ®'l' and the number of ordinary irreduciblecharacters in B rr is given by I(A)~). Hence we have~~m(n) = ~ s(n- ;'P)l(i..).A=OLemma 1.i\leA) - 1*(J.) -= :E 1*(J. - {1)m({j).~=l .1) . For the notion of hooks, see T. Kakaynma, Oil some modular properties of theirreducible represel1tations of symmetric gloups 1. H, Jap. J. ~lath. 17 (19-11'): we referto these papers as NI and NfI.2) See NIL p.413.3) R. Brauer, On a cOl1jecture by Nakayama, Trans. Ray. Soc. Canada, 41 (1947).4) T. Nakayuma and M. Osima, Note Oil blocks of symmetric groups, }.jagoya 1\1ath.J. 2 (1951).631Osima: On some character relations of symmetric groupsProduced by The Berkeley Electronic Press, 195264 MA-SA-HU OSUfAProof. From our definitionlI.1(J.) = b l*(). - J.tl)m().p).\p=olI.1*(J.) + ~ 1*(J. - J.p)m(J.p)•1I.P-1Let Cl' C2 , •••••• , Cm(n; be the classes of conjugate elements in@5,I. If ClI contains an element G such that G is a permutation com-posed of cycles of lengths al' a 2 , •••••• , a", (a\ ?; «2 ~ .•..•• ? a h > O)~then ClI is characterized by a partition (aJ. Hence we denote C'IJ by. C(aJ.Lem~a 2. The number of classes C(a l ) in ®kP such that at = {11P(i = 1,2, .... :.) is equal to ni(k).Proof. Every C({1,P) determines uniquely a class C«(jl) in 0},~ ~and conversely.A p-regular element of ®1/ is an element whose order is primeto p; the other elements are said to be p-singular. Similarly, wedenote the classes of conjugate elements as p-regular or p-singularaccording as the elements of the classes are p-regular or p-singular.Let us denote by m*(n) the number of p-regular classes in ®1l. Thenwe haveLemma 3.11:m(n) - m*(n) = ~ m*(n - f3P)m«(j).13-1Proof. If C(a;) is a p-singular class, then at least one at isdivisible by p. LetPI ~ {32 ~ •••••• > 8t > 0 and the remaining at be prime to p. Sucha/ we denote ,by rI' r2, , rh-t:(r j' P) = 1.If ~ f3 l = f3; then C(a[) determines uniquely C(8t ) in ®{J and' p-regularC(rJ) in ®n -(JtJ • Since the converse is also valid, we obtain ourassertion by Lemma 2.Theorem 1. Let m*(n) be the number of p-regular classes in @S,l.kThen 1n*(n) = ~ s(n - ).P)l*().).lI.=1IProof. Our assertion is evidently valid when k = 0, that is,.n < p. Let k > 0 and assume that the theorem is true for ®n -(Jl).([1 = 1, 2, , k). Then we have2Mathematical Journal of Okayama University, Vol. 1 [1952], Iss. 1, Art. 3http://escholarship.lib.okayama-u.ac.jp/mjou/vol1/iss1/3ON S03fE CHARACTER RELATIONS OF SYMMETRIC GHOUPS GSk-$ km*(n - #P) = h s(n - ({3 + (1)P)I*(C1) = ~ s(n - J.P)I*(J. - (j).u=o A=~Hence it follows from Lemma 3 thatmen) - m*(n)k k= ~ (~ s(n - J.P)I* (A - (1»1n«(3){3=l :\={3k Ab (~ 1* (J. - l3)m({j»s(n - i.P)A=1 13=1Tt'~ (l().) - 1*(J.))s(n - J.P):\=1Tt;= men) - h s(n - ;'P)l*(i.)..\=110;Whence we have m*(n) = ~ stn - ;'P)l*(i.).'\=12. Let Xl' X2, •••••• , Xm(n) be the distinct ordinary irreduciblecharacters of ®'l' Let us denote by G(G) a class of conjugate ele-ments which contains an element G. Since G-1 E G(G), we have(6) i = 1, 2, , men).From the orthogonality relations for ordinary group characters, wehave( 7)for G(G/Ao) = G(G)for G(G/Ao) =i= G(G,,)where n(G/Ao) is the order of the normalizer N(G,J. If V is anyjJ-regular element of ®,p then among In(n) characters X1(V), X2(V),.••••. ,Xnt(n)(V), there exist m*(n) linearly independent Xj(V), In the:following we shall determine all linear relations between Xl(V), x~(V),.••• •.. ,. Xntcn)(V) by Murnaghan's recurrence rule l ).Murnaghan's recurrence rule. Let Hi' ~, be the totality ofg-hooks in the diagram T, and let r v be the height of H)). If Q" is anelement of ®n containing a g-cycle and if Q is the permutation ofn - g letters obtained from Q by removing this cycle, thenx(T: Q) = (-lYl-1x(T-H1 : Q)+ (-lY~-lx(T-H~: Q) + .1) F. D. l\!urllaghan, On the representations 01 the symmetric group, Amer. J'," ~lath~59 (l9~7). Cf. also NI, Appendix.3Osima: On some character relations of symmetric groupsProduced by The Berkeley Electronic Press, 195266 lIASAHU OSI1\IAwhere x(T), x(T - Hy ) denote the characters belonging to the diagramsT, T - H',i. If T possesses no g-hook, then x(T: Q) = o.Let us denote by Q the elements of en containing at least oneg-cycle. Then there exist m(n - g) classes C(Ql)' C(Q2), •••••. ,(C Qm~1l-9»which contain the elements Q. If (J.., is the permutation of 11, - gletters obtained from Qy by removing a g-cycle, then C«(J1)' C(Q2),.•.... , C(Qm~n-I)) are all the classes of conjugate elements in en-I).From (6), we have(8 )men)1t:t x/(U)x,(Q,)men)~ Xt(Q/L)Xi(QlI)i-I.owhere U is any element of en without g-cycle. Applying the re-currence rule to Xi(Q,J in (8). we obtain( 9 )m(n-u) _{~ RiXt( U»xr(Q..,> = 0j-1men-g) _2j RixL(Q/L»)r.1(Q·J - n(Q/L)o/LV)=1where x1 (j = 1, 2, ...... , m(n - g» are the irreducible characters of@5n-v and R;(Xl(G» for any G E en is a linear combination of X1(G),X2( G), ...... , xm(u)(G). Since xt, x:, ...... , X:(n-u) are linearly independent,we have from the first formula (9)(10) Rj(X'i(U» = 0 j = 1, 2, ...... , m(n - g)for all elements U without g:cycle.Lemma 4. Let T* be a diagram of en-go If T* contains p(r)g-hooks of the same height r, then we can obtain p(r) + 1 distinct dia-grams of ®n by adjoining a g-hook of the height r to T*.Proof. When p(r) = 0, our assertion is valid by T. Nakayamal).Hence, by induction with respect to p(r), we can show that ourassertion is true for any p(r).Let Tt be the diagram of @5n-o corresponding to x1, and letTt~, TJ~~, ...... , Tt~r)+lbe the diagram of @:in obtained from Tf by adjoining a g-hook of the1) See NIL p.114.4Mathematical Journal of Okayama University, Vol. 1 [1952], Iss. 1, Art. 3http://escholarship.lib.okayama-u.ac.jp/mjou/vol1/iss1/3OX SOME CHARACTER RELATIONS OF SY:\Il\IETR1C GROUPS 67height r. If we denote by X)~)tr the irreducible character belonging toTt~, then we can see thatfJ p(r)+l(11) Rixi(G» = b ~ (-IY-lx)~)tr(G) (for all G E €Ill).r-l (T~lTheorem 2. R 1(Xf(G», R 2(Xi(G», .•. "', Rmcn-fJ)(Xl(G» are linearlyindependent.Proof. If we set(j row index, p. column index: j, p. = 1, 2, ...... , m(n - g», then thesecond formula (9) becomes2'M = (n(QJJ.)8JJ.'.) = D.Since D is non-singular, we have I M I =l= o. Hence R 1(Xi(QJJ.»'R 2(Xt(QJJ.» , ••. "', Rm(n-fJ)(Xl(QJJ.» (p = 1, 2, ... "', 1n(n - g» are linearlyindependent. This fact shows that the theorem is valid.If -we put, in particular, g =J.p (.< =1, 2, ... "', k) in (10), thenwe obtain(12) j = 1, 2, , m(n - ;,P), .t'= 1, 2" "', k.'where V is any p-regular element 'of ®Il.10:Lemma 5. 1n(n) - m*(n) :::;;: b m(n - J.P).A~lFor the sake of simplicity, we set u = m(n) -m*(n) and v =k~ m(n - ;'P). Let us denote by11.-1(13)the p-singular classes in ®ll such that PSA) contains a ,<p-cycle butdoes not contain a ).Ip-cycle (J. < J./). Then C (PJ>") (/1. = 1, 2, ,d(A.), J. = 1, 2, , k) give all the p-singular classes in ~n. Hence(14)kU = ~ d(J.).A~lLet PJA.) be an element of ®1l-A21 obtained from PJl\) by removing a;.p-cycle. Then, similarly as (9), we have1) By a matrix of type (a, b) we understand a matrix witlr a rows and &columns.5Osima: On some character relations of symmetric groupsProduced by The Berkeley Electronic Press, 195268(15)If we setl\{A8ARU OSIMA(m(~p)RY')(Xl(P~>'»)X}>')(J5S>'» =j-1m~n->.p) _~ R?)(Xi(P~IC»)X)A)(PS>'» = 0j==1(j row index, I-t, 11 column index: j = 1, 2, ... "', m(n - J.P), 11 = 1, 2,....•. , d(K), p. = 1, 2, ,..... , d().», then (15) becomes(16)Hence we have{Z~M>.>. = (n(P~>'»ofLY) = DJ.Z~M>.IC = O.(Z; ) (MU Ml2 MU:) (DlZ( '. .:~.~'~.::::::.~~ = D,.Z~ \Mkl M k2 •••••• M"kSince DJ. (J. := 1, 2, .. ,.,., k) are non-singular, the matrix (~J.) (K, J. =k1, 2, ... "', k) which is of type (v, u) has a rank u = ~ d(J.). This>'=1implies that there exist u linearly independent R~>')(xi.(P» among vRY)(Xl(P» where P is any p-singular element of en' This fact, com-bined with (12), shows that if R(Xl(V» = ~ alxi(V) = 0 for all p-regularelements V, then R(Xl(G» (for any G E en) is a linear combination ofRjA)(Xl(G».The relations (12) seem to be useful to determine the irreduciblemodular characters of @5n, but we have only succeeded to determinethe characters belonging to the P-blocks of next-highest kind.In the forthcoming paper, we shall study the properties ofRj>-)(Xi(G» in detail.DEPARTMENT OF MATHEMATIC~,OKAYAMA- UNIVERSITY(Recez'ved January 10, 195'1)6Mathematical Journal of Okayama University, Vol. 1 [1952], Iss. 1, Art. 3http://escholarship.lib.okayama-u.ac.jp/mjou/vol1/iss1/3

On some character relations of symmetric groups

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