SOME CHARACTERS OF THE SYMMETRIC GROUP R. E. INGRAM, SJ.

Transcription

1 SOME CHARACTERS OF THE SYMMETRIC GROUP R. E. INGRAM, SJ. Introducton. Frobenus []1 derved expressons for the characters of a few very smpe casses of Sm, the symmetrc group on m thngs. Here we gve formuas for some more compcated casses. The method s qute genera. A recurrence formua, due to Murnaghan, s used. The need for the formuas arose from genera consderatons of nucear bndng forces.2 Some appcatons w be gven n another paper. 1. The character x(p)x, of the cass ax = m p, ap = [, 2]. If {X}, n the usua notaton, s a partton of m, the same partton may aso be represented [6] by two sets of numbers {b, a] and the foowng reaton hods, (1.1) (X,- - /)" = >" + (~ D"+1 + 1)" + (- 1)" /. 1=1 j- - - If xx s the character of a cass K of Sm, {x/} = {X, X2, X,- X*} a partton om p, and xx; the character of the cass K' of Sm-P, where K' has the same cycc structure as K, but wth one ess />-cyce, then (1.2) xx = Xx' (Murnaghan Recurrence Formua [6]). - Denotng by x( )x the character of the cass havng one -cyce (a others unary), we have where D s the dmenson ' (m-p)m) h\h\ (h - p)\ h' k x(p)\ = E Xx)- = E D\) of the representaton. k U() = n (/, - y, p = x* + k - p) Receved by the edtors Apr 24, Numbers n brackets refer to bbography at end of ths paper. 2 The orgna probem to fnd \{2, 2)\ s due to Dr. J. A. Wheeer, Prnceton [7]. 358 Lcense or copyrght restrctons may appy to redstrbuton; see

2 SOME CHARACTERS OF THE SYMMETRIC GROUP 359 O - PV- h- fh - P) m (h-p)\ -pfd) x t (where /(*) - J[ (* ~ f(x - pn fx) J^,. au»- d-- (x-p + ) J \. Ths expresson for D\: may be wrtten as Z* Aj/x \,) pus a poynoma n x, where Aj = jj 1) j p + Y)fj p)/f'{j). If we expand Z* Aj/x f) n descendng powers of x, t s evdent that y?.4/ s the coeffcent of 1/x n the expanson of * fx - p) (1.3) *(* -)-..(» ~f+) fx) n descendng powers of x. Lkewse (1.4) Z,v4,- s the coeffcent of 1/x2 and k a +1 (1.5) Z IjAj s the coeffcent of 1/x Thus and takng a = 0, we have3 * a m- p 1 * a Z^;.=-fA-EM * m - p /-1\ * (1.6) 1 *' W ' 1 -^^ = rx(x-)..-(x-p+)^^. It shoud be noted that f {X/ } ends n a negatve number when arranged n non-ascendng order [6], then Kj p+k j <0, asx, p may be ncreased by k j by movng down to kth poston from jth poston. Therefore I, - p < 0, * [ sgnfes the coeffcent of 1/x n the expanson of the functon wthn the brackets. Lcense or copyrght restrctons may appy to redstrbuton; see

3 360 R.. INGRAM (June // - P + 1 ^ 0. Therefore A,- = 0. If cannot be rearranged to form a non-ncreasng sequence X,- p = X,-+, S, some S, X, + j - K - p = X,+. +j- K-S, /, /> = /,+,. Therefore In nether case s any contrbuton Let x=y+k', then (1.7) where r fh - P) = 0, A = 0.»(»- ) (* - p +1) ^ - made to J ^4y. /(*- p) L /(*) J y. y(y - 1) (y - P + D [F(y f(y) - py Jv. and f(y)-n[, ^"^J» a, 6 as n (1.1), f-l(y + <*/ + 1)J F(y P) wt2 1 - />ny) y r 1/3 /> \ (1.8) + (c5 + 2pa - pmct + 2p*c3 - phn* + pm) 7* 1 / p I--I a-cs - pmc + pct p*mc» y*\ p*m* />Jc4 + - m*p* + />3c, + m/>«) / Lcense or copyrght restrctons may appy to redstrbuton; see

4 1950] SOME CHARACTERS OF THE SYMMETRIC GROUP 361 wth «= E + 1)}. ct,+ = {**'- (<*; +1)"}. 1 ^ = r{*r+(a/+),r}. Usng (1.7) and (1.8) n (1.6), we get the expressons (5.1), (5.2), (5.3), (5.4), (5.5) on pp The character x(p- 2)x of the cass a = m p 2, aj=, orp =. Let x(/>, 2)x be the character of the cass at = 1 bnary cyce and ap = 1 cyce on etters. By the recurrence formua, (1.2), (2.1) x(p, 2)x = E x(2)xj ;- where X/, as a subscrpt, ndcates that x(2) s the character of the representaton {X/ } = {X, X2,, Xy p,, X*} of the symmetrc group on m p etters, Sm-P, correspondng to the cass ot =m p 2 unary cyces, a2 = bnary cyce. By (5.2), X(2)x'y = {MM A,(m -p-2)\/(m-p)\ where (Af2)/ s Mt wth Xy p nstead of Xy (that s, the correct vaue of M for {Xy}). D\'s s the dmenson of {X/ }, and m p repaces m as {X/ } s a representaton of Sm-pm* = Z(x,-)(Xy-4-D- /' + E j- s a functon of X, Xj,, X*. (Mt)I s the vaue for X, X2,, (Xy p),, X*, Therefore Then, (2.1), (M)/ - Mt - 2p{\, - j) + p(p - 1). Mt + P(P - 1) - 2#(X, - j) (w - p)(m - p - 1) Lcense or copyrght restrctons may appy to redstrbuton; see

5 362 R. E. INGRAM [June * M ) - 2#(X, - 7) xo. 2)x = 2--;-^7-7 r:-a; (w p)(m p 1) f ) * (2-2) - f -' E Dy. (m p){m p 1) E (x.- S)h (m p)(m p 1) (snce Af2, p, w are the same n a terms). x(p)x = E' A has aready been found. E (x,- = E - k k where * m - p I 1 * \ E,Dy. = --f- /A - - E,A t) m X p % / L /(*) J/x«(that s, the coeffcent of 1/x2 n the expanson of x(x ) (x-p + )f(x-p)/f{x)). Equvaenty EM/= [*»(*- 1) (*-# +1)^7T^1 L f(«) J /. Puttng, as n (1.7), a: = y+, we have r /(* - /O L /(*) J /x Therefore = ~(y + *)y(y- ) (y- /> + ) ^"^, L F(y) J /y * T F(y - P) m-p E *V = yy ) ' y P + D, ^ A. L - pf(y)aw m h (2.3) E (x,- = E *A} - * E A} Lcense or copyrght restrctons may appy to redstrbuton; see

6 SOME CHARACTERS OF THE SYMMETRIC GROUP 363 = \\(y +k)y-(y-p+ )Hy P)] (L F(y) J /u (2.3) - k \y (y - p + 1) V- Dx L F(y) *"(y) J / /y j) m F(y - p)~\ m p Ih. pf(y) Jv» m From the genera expresson (2.2) the formuas for x(2, 2), x(3, 2), x(4, 2) are obtaned, for exampe formua for x(2, 2) (that s, p = 2): By (2.2), (AT, + 2) * 4 * (m 2){m 3) (w 2)(m 3) x(2' 2)x =r-5v-t Z A; - r(-r Z (Xy- M*. ZA; = x(2)x = - w(w 1) * Dx r F(y - 2)" m(f» 1) L 2F(y) J V, Usng (1.3) wth p 2, mutpyng by y2(y 1), and takng the coeffcent of 1/y, we have hence X _ >x(c4 + 2c3 +2m- m2) '* " m(m - 1)(«- 2)(» - 3) (Af 24-2)M2-4(c4 + 3c3 4-4m - w2) 4-4(c3 4-2w) x(2' 2)x =-?-7Tr- -«- w(m - )(m - 2)(w - 3) A (2.4) 2 M 2Af 34-4w(w - 1) =- D\ m(m \)(m 2)(m 3) where M2, Af3 are as defned prevousy. 3. The character %(P> 3). Let x(p, 3) be the character of the cass a = m p 3 unary cyces, a3 = ternary cyce (cyce on 3 etters) and ap = 1 cyce on p etters. By the recurrence formua (1.2), (3-D xp- 3) = Z x(3)x; Lcense or copyrght restrctons may appy to redstrbuton; see

7 364 R. E. INGRAM [June where X/ as a subscrpt, ndcates that x(3) s the character of a representaton {X/ } = {X, X2,, (X, p),, X*} of Sm-P correspondng to the cass obtaned from the orgna cass by droppng the cyce on ^-etters; t s, then, the character of a cass wth one ternary cyce and a other cyces unary. x(3)xy' s known, t s gven by 1, cf. (5.2) x(3)x/ = [/2(Jf3)/ - 3/2(m - p)(m -p- 1)] " ~ * ~ 3' L\, m p 1 where (M3)j s the same as M% but has Xy # n pace of Xy, m s repaced by m p as {X/ } s a representaton of Sm-V, and D\s s the dmenson of {X/ }. Mt = z [2(Xy- j)«4-3(xy - )' + (Xy- j) + 2j* - 3^ + j] 1 s a functon of X, X2,, X*. (Jf,)y s found by substtutng for Xy, Xy # (Jf,)/ = If, - 6#(Xy - j)2 + 6p(p - )(Xy - ) - p\2p - 3). X(3)xj = [1/2Jf, - 3#(X, " )2 + 3y(> - )(Xy - j) - p*/2(2p - 3) and by (3.1) (m p 3)! - 3/2(» -p)(m-p- )]/Aj.-^ m p x(#, 3)x = E [y ^3-3#(X, - j)2 4-3p(p - )(Xy - j) P* 3 "I (m - p - 3)! - ^- (2# - 3) - (m - p)(m - f - 1) Z>x; _]»» #! (3.2) = [- Jf, - #»/2(2* - 3) 3 1 (m - * - 3)! * _ (m ^ _ ) 2- - EAj 2 J w #! ' * r, w - p p z [(Xy - )2 - (p - )(Xy - 1)]-V ~ Dyr 1 wt p Zj= Ay = x(p) and s known by the resuts of 1. Lcense or copyrght restrctons may appy to redstrbuton; see

8 95 ] SOME CHARACTERS OF THE SYMMETRIC GROUP 365 z [(Xz-jr-o D(x,-)fxj where \j j = j k as prevousy n 1. = S (*J - *)2- (P- Oft " *)]A} Z & - k)l\, T F(y - Ä)" m - a! (3.3) ky-d-o-h).,p/, -r A, L - pf(y) J/y m * 2 m- p / * 2 \ as n 1, where 4y=/,(/,-1) (h-p+)f(h-p)/f'() and /(*) = I and Z = [*(* L - I) (*- p + 1) ^'"/H /(*) J,z* = r*»(«-1). o - *+1)/(* ~p)) L /(*) J u, = "(y + kyy{y - 1) (y- f + ^1, Z (h - k)u, 2*(y + *) + *2}y(y - 1) = \f(y-d---(y-p+)hy~p)'], L F(y) J v«z (Xy- = ~y3(y - 1) (y- P + 1) ^"^ L - pf(y) J/ w --öx- m ny) J/» From these expressons, x(p> 3)x s obtaned. (If p = 2 the character x(2, 3) s obtaned, t s of course the same expresson as n 2.) Lcense or copyrght restrctons may appy to redstrbuton; see

9 366 R. E. INGRAM [June Takng p = 3, we have, m 6! * x(3, 3)x = [/2M3-27/2-3/2(m - 3){m - 4)]- A', w 3! *.. m 61-9 E [(Xy - j)» - 2(XJ- j)]dyj - m 31 [1/2M3-3/2m(m - 1)]L\ ZA, = x(3) m(m )(w 2) E [(X3-)2-2(X) -)]»x. Dx T /?(y - 3)n =- y\y - )(y - 2)2^m(m - 1)(» - 2) L - 3F{y)\ Usng the expanson of F(y 3)/ 3F(y) gven by (1.8) wth p = 3, mutped by y2(y )(y 2)2, and coectng the coeffcent of 1/y, fnay we obtan the expresson for x(3, 3), x(3, 3) = [/4Jfj - 18M6 4-54f3-6(m - 13m - 39)M3 wth M, M3, M as defned prevousy. 2 m 6! 4-9m(m - )(m - 13m 4-34)]Z?X -> m 4. The character x(2, 2, 2)\. It s evdent that, by means of the recurrence formua, formuas can be found for characters of casses consstng of three non-unary cyces. The expressons nvoved, however, become too compcated for easy computaton. To ustrate the method t s suffcent to take the smpest case, the cass consstng of three bnary cyces and m 6 unary cyces. By the recurrence formua (1.2), x(2, 2, 2)x = E X(22V j- where x(2, 2, 2)\ s the requred character of the representaton {X} of Sm, x(2, 2)\j- s the character of the cass, consstng of 2 bnary cyces and m 6 unary cyces, of Sm-, and {\/ } s the representaton {X, X2,, Xy 2,, Xt} of Sm-. x(2, 2)x'j s known, t s gven by (2.3) or (5.5). Lcense or copyrght restrctons may appy to redstrbuton; see

10 1950] SOME CHARACTERS OF THE SYMMETRIC GROUP 367 * [(Af2)/ ]2-2(3/,)/ + 4(m - 2)(m - 3) X( ' h> J (m - 2)(» - 3)(» - 4)(» - 5) (M)j s Af2 wrtten for X/, that s, Xy 2 s wrtten for Xy, as M2 = ZJ Xy(Xy-2j+). [(M)! f = M - 8(Xy - j)m + AM + 16(Xy - /)" - 16(Xy - j) 4-4, (J/,)/ = M3-12(X, - )2 + 12(Xy - ) - 6, (4.1) *. 2 '«61 x(2, 2, 2) = [Af 2 + 4Af 2-2M, 4-4w - 20m 4-40]ZV.- m 2! * r, w 6! + [40(Xy - )2-8(M2+ 5)(Xy - j)l\;] -- m 2! The frst of these two sums s [M\-\--\M2 2Ms-\-Am 20m -\--O]MDx(m 6)\/m. The second s obtaned by the same reasonng as n the prevous sectons, Z [40(Xy - j)2-8(m2 + 5)(Xy - /)]&- r F(y - 2)1 m 6! = (40y2-8(2/, 4-5)y)y(y - 1) - TV- L - 2F(y)Ay J y m r F(y - 2)1 w - 6! = 8y(y -M- 5)y(y - 1) - 7A- L 2F(y) J / m! The expanson of F(y 2)/ 2F(y) s found from (1.8), and takng the coeffcent of 1/y when ths seres s mutped by 8y2(y 1) (y M 5), we get r 3 2, m 6! x(2,2,2)x = [M - 6M2M3 + 40Af4 + 12(m - 9m + 10)M2] >x- m 5. Resuts. x(2)x m (5.1) -= M2 (Frobenus), D\ m 21 x(3)x m Ms (5.2)-=-3/2m(m - 1) (Frobenus), Dx m 31 2 x(4)x m! (5.3)-= Ma - 2(2m - 3)Af2 (Frobenus), Z>x w 4! Lcense or copyrght restrctons may appy to redstrbuton; see

11 368 R. E. INGRAM [June x(5)x m Jf f 49-15»» 2 -=-4--M3-5/2Jf2 (5.4) Dx m /6m(m - )(5m - 19), x(2, 2)x m 2 (5.5)-= Jf, - 2Jf, 4-4*»(w - 1), Dx m- 41 where x(3, 2)x m! = 1 Dx»-5! 2 [M,Jf, - 6Jf«- 3/2(V - Urn 4-16)Jf,], x(4, 2)x m, -:-= Jf2Jf4-4Jfs - 2(2w - )Jf, D\ m 6! 4-4/3(12»» - 34)Jf, - 32/3m(m - I)(2m - 7), x(3, 3)x m = Jf, - 18Jf s 4-54Af, Dx m - 6! 4-6(w2-13m - 39)Ms + 9m(m - )(m - 13m + 34)], x(2,2, 2)»»!, Jf, - 6Jf,Jf, 4-40Jf4 Dx m - 6! Mt= - 1)1 + 12(m - 9m+ 10)Jf, = E [(X*- 7)(X, ) - 7(7 " DJ. Jf, = E [*»(»/ + )(26y 4-1) 4- a,(a + )(2a, 4-1)J = [(X,- )(X, -j+ 1)(2X, ) 4- j(j - D(2j - 1)1, m«= bvb+d* 1)*] 1 - [(Xy - 7)'(X, - / 4- Ds - fu ~ D*. Lcense or copyrght restrctons may appy to redstrbuton; see

12 some characters of the symmetrc group 369 Mt = [b)(jb, + )\2b,- + 1) + aaf + )\2af + 1)] = Z [0/ - y)j(x,- j + 1)S(2X,- - 2j + 1) + sc - OT/ - DJ, w!a(7) ma(b)a(a) L\ =-=kk h A A A n>,! n«,!t o, +<*,+1),«= Seected Bbography 1. G. Frobenus, Über de Charaktere der symmetrschen Gruppe, Berner Berchte (1901). 2. -, Über de charakterstschen Enheten der symmetrschen Gruppe, Berner Berchte (1903). 3. T. Schur, Über de Darsteung der symmetrschen Gruppe durch neare homogene Substtutonen, Berner Berchte (1908). 4. F. D. Murnaghan, On the representatons of the symmetrc group, Amer. J. Math. vo. 59 (1937). 5. -, The characters of the symmetrc group, Amer. J. Math. vo. 59 (1937). 6. -, The theory of group representatons, Johns Hopkns Press. 7. J. A. Wheeer, Vewponts n nucear structure, Physcs Revew vo. 52 (1937). 8. E. Wgner, () Symmetry of nucear hamtonan, Physcs Revew vo. 52 (1937). 9. -, () On the saturaton of exchange forces, Proc. Nat. Acad. Se. U.S.A. vo. 22 (1936) p H. Wey, The cassca groups, Prnceton Unversty Press, , Theory of groups and quantum mechancs, Dutton. 12. G. de B. Robnson, On the representatons of the symmetrc group, Amer. J. Math. vo. 60 (1938) pp , On the representatons of the symmetrc group, Amer. J. Math. vo. 60 (1938). 14. T. Nakayama, On some moduar propertes of rreducbe representatons of a symmetrc group, Part I, Jap. J. Math. vo. 17 (1940); Part II, bd. 15. D. E. Lttewood, The theory of group characters, Oxford, Unversty of Ireand Lcense or copyrght restrctons may appy to redstrbuton; see