Notes on PSL(2, 11) I

Transcription

1 Notes on PSL(2, 11) I Brian G Wybourne % Instytut Fizyki, Uniwersytet Miko laja Kopernika, ul. Grudzi adzka 5/7, Toruń, Poland 4 July 2003 Abstract. These notes are compiled as my attempt to understand some of the properties of the embedding of the icosahedral group, I, in the finite group PSL(2, 11). % bgw@phys.uni.torun.pl WEB: bgw 1

2 2 Brian G Wybourne 1. Introduction These notes are compiled as my attempt to understand some of the properties of the embedding of the icosahedral group in the finite group PSL(2, 11) 1.1. Characters of P SL(2, 11) Table 1 The Characters ofpsl(2, 11). (1 11 ) 55( ) 110( ) 132(5 2 1) + 132(5 2 1) 110(632) 60(11 ) + 60(11 ) χ χ β β χ β β χ χ χ χ α α χ α α α = 1 2 ( 1 + 5), α = 1 2 (1 + 5), β = 1 2 ( 1 + i 11) NB. We have used the partitions appropriate to the classes of the symmetric group S 11. The group PSL(2, 11) is a subgroup of S 11. Note that the conjugacy classes all derive from even permutations which are appropriate to the group A Characters of the Icosahedral Group Chemists commonly present the character table as below Table 2 Characters of the ordinary irreducible representations of I. 1 20C 3 15C 2 12C 5 12C5 2 A T ( ) (1 5) 2 2 T (1 1 5) (1 + 5) 2 2 U V while in terms of the isomorphic alternating group A 5 they can be written as Table 3 Characters of the ordinary irreducible representations of A 5. (1 5 ) 20(31 2 ) 15(2 2 1) 12(5) + 12(5) [5] [ ] ( ) (1 5) 2 2 [31 2 ] (1 1 5) (1 + 5) 2 2 [41] [32]

3 Notes on PSL(2, 11) I 3 NB. We designate irreducible representations of A n by partitions of n enclosed in square brackets, [, ], and distinguish splitting irreducible representations by attaching a ± as a subscript The P SL(2, 11) I Branching Rules These follow from the character tables of the respective groups to give Table 4 The P SL(2, 11) I Branching Rules. χ 1 A χ 2 V χ 3 V χ 4 T 1 + T 2 + U χ 5 A + U + V χ 6 T 1 + T 2 + V χ 7 T 1 + U + V χ 8 T 2 + U + V 1.4. Kronecker Products for Ordinary Representations of I Table 5 Kronecker products for the ordinary irreducible representations of I. A T 1 T 2 U V A A T 1 T 2 U V T 1 T 1 {A + V } + [T 1 ] U + V T 2 + U + V T 1 + T 2 + U + V T 2 T 2 U + V {A + V } + [T 2 ] T 1 + U + V T 1 + T 2 + U + V U U T 2 + U + V T 1 + U + V {A + U + V } + [T 1 + T 2 ] T 1 + T 2 + U + 2V V V T 1 + T 2 + U + V T 1 + T 2 + U + V T 1 + T 2 + U + 2V {A + U + 2V } + [T 1 + T 2 + U] 1.5. Kronecker Squares for the representations of PSL(2, 11) Below we give the resolution of the Kronecker squares of the irreducible representations of PSL(2, 11) into their symmetric and antisymmetric terms. Table 6 Resolution of the Kronecker squares of the irreducible representations of PSL(2, 11). χ i χ i {2} χ i {1 2 } χ 1 χ 1 χ 2 χ 3 + χ 5 χ 4 χ 3 χ 2 + χ 5 χ 4 χ 4 χ 1 + χ 2 + χ 3 + 2χ 5 + χ 7 + χ 8 χ 4 + χ 6 + χ 7 + χ 8 χ 5 χ 1 + χ 2 + χ 3 + 2χ 5 + χ 7 + χ 8 χ 4 + χ 6 + χ 7 + χ 8 χ 6 χ 1 + χ 2 + χ 3 + 2χ 5 + χ 6 + χ 7 + χ 8 2χ 4 + χ 6 + χ 7 + χ 8 χ 7 χ 1 + χ 2 + χ 3 + 2χ 5 + χ 6 + χ 7 + 2χ 8 2χ 4 + 2χ 6 + χ 7 + χ 8 χ 8 χ 1 + χ 2 + χ 3 + 2χ 5 + χ 6 + 2χ 7 + χ 8 2χ 4 + 2χ 6 + χ 7 + χ 8

4 4 Brian G Wybourne Table 6a Resolution of non-trivial Kronecker products of irreducible representations of PSL(2, 11). χ 2 χ 3 = χ 1 + χ 7 + χ 8 χ 2 χ 4 = χ 3 + χ 4 + χ 6 + χ 7 + χ 8 χ 2 χ 5 = χ 3 + χ 5 + χ 6 + χ 7 + χ 8 χ 2 χ 6 = χ 4 + χ 5 + χ 6 + χ 7 + χ 8 χ 2 χ 7 = χ 2 + χ 4 + χ 5 + χ 6 + χ 7 + χ 8 χ 2 χ 8 = χ 2 + χ 4 + χ 5 + χ 6 + χ 7 + χ 8 χ 3 χ 4 = χ 2 + χ 4 + χ 6 + χ 7 + χ 8 χ 3 χ 5 = χ 2 + χ 5 + χ 6 + χ 7 + χ 8 χ 3 χ 6 = χ 4 + χ 5 + χ 6 + χ 7 + χ 8 χ 3 χ 7 = χ 3 + χ 4 + χ 5 + χ 6 + χ 7 + χ 8 χ 3 χ 8 = χ 3 + χ 4 + χ 5 + χ 6 + χ 7 + χ 8 χ 4 χ 5 = 2χ 4 + χ 5 + 2χ 6 + 2χ 7 + 2χ 8 χ 4 χ 6 = χ 2 + χ 3 + χ 4 + 2χ 5 + 2χ 6 + 2χ 7 + 2χ 8 χ 4 χ 7 = χ 2 + χ 3 + 2χ 4 + 2χ 5 + 2χ 6 + 2χ 7 + 2χ 8 χ 4 χ 8 = χ 2 + χ 3 + 2χ 4 + 2χ 5 + 2χ 6 + 2χ 7 + 2χ 8 χ 5 χ 6 = χ 2 + χ 3 + 2χ 4 + χ 5 + 2χ 6 + 2χ 7 + 2χ 8 χ 5 χ 7 = χ 2 + χ 3 + 2χ 4 + 2χ 5 + 2χ 6 + 2χ 7 + 2χ 8 χ 5 χ 8 = χ 2 + χ 3 + 2χ 4 + 2χ 5 + 2χ 6 + 2χ 7 + 2χ 8 χ 6 χ 7 = χ 2 + χ 3 + 2χ 4 + 2χ 5 + 2χ 6 + 3χ 7 + 2χ 8 χ 6 χ 8 = χ 2 + χ 3 + 2χ 4 + 2χ 5 + 2χ 6 + 2χ 7 + 3χ 8 χ 7 χ 8 = χ 2 + χ 3 + 2χ 4 + 2χ 5 + 2χ 6 + 3χ 7 + 3χ 8 Inspection of the above results allows us to conclude that the irreps χ 2, χ 3 form a complex pair while the remaining irreducible representations are all real and orthogonal Embedding PSL(2, 11) in S 11 The irreducible representation {10 1} of the symmetric group S 11 is of dimension 10 which is real and orthogonal, though NOT unimodular. We note that the symmetrized square of this irreducible representation may be resolved as {10 1} {2} = {11 } + {10 1} + {92} (1) {10 1} {1 2 } = {91 2 } (2) Three possible embeddings in {10 1} might be considered {10 1} χ 2 + χ 3 (3a) {10 1} χ 4 (3b) {10 1} χ 5 (3c)

5 Notes on PSL(2, 11) I 5 The embedding (3b) is inconsistent with (1) since the symmetric square of χ 4 does not contain itself. In the case of (3a) we are immediately led to the decompositions {11 } χ 1 (4a) {10 1} χ 2 + χ 3 (4b) {92} 2χ 5 + χ 7 + χ 8 (4c) {91 2 } χ 1 + 2χ 4 + χ 7 + χ 8 (4d) In the case of (3c) we are led to the decompositions {11 } χ 1 (5a) {10 1} χ 5 (5b) {92} χ 2 + χ 3 + χ 5 + χ 7 + χ 8 (5c) {91 2 } χ 4 + χ 6 + χ 7 + χ 8 (5d) These branching rules can be readily extended by using the S 11 characters evaluated over the classes of PSL(2, 11) S 11 PSL(2, 11) Branching Rules Below we give the relevant S 11 PSL(2, 11) decompositions for the embedding defined by (3c). Since under S 11 PSL(2, 11) the decomposition of the irreducible representations labelled by conjugate partitions are the same we give only the decomposition in the case of only one partition of each conjugate pair. Note that in S 11 the two irreducible representations {61 5 } and {43 2 1} are self-conjugate. Recall that under S n A n the character associated with a self-conjugate irreducible representation of S n is the sum of two simple characters of A n. These characters of A n take exactly half the values of the characteristics of S n save for the splitting classes of A n. In the case of A 11 the splitting classes are (11 ) ± and (731) ±.

6 6 Brian G Wybourne Table 7 Branching rules for S 11 PSL(2, 11). Dim(λ) {λ} Decomposition 1 {11} χ 1 10 {10 1} χ 5 44 {92} χ 2 + χ 3 + χ 5 + χ 7 + χ 8 45 {91 2 } χ 4 + χ 6 + χ 7 + χ {83} χ 1 + χ 2 + χ 3 + χ 4 + 3χ 5 + χ 6 + 2χ 7 + 2χ {821} 2χ 2 + 2χ 3 + 3χ 4 + 3χ 5 + 5χ 6 + 4χ 7 + 4χ {81 3 } 4χ 4 + χ 5 + 2χ 6 + 2χ 7 + 2χ {74} χ 1 + χ 2 + χ 3 + 2χ 4 + 4χ 5 + 2χ 6 + 3χ 7 + 3χ {731} χ 1 + 4χ 2 + 4χ 3 + 9χ 4 + 8χ 5 + 9χ χ χ {72 2 } χ 1 + 4χ 2 + 4χ 3 + 4χ 4 + 7χ 5 + 6χ 6 + 7χ 7 + 7χ {721 2 } 4χ 2 + 4χ χ 4 + 8χ χ χ χ {71 4 } χ 1 + χ 2 + χ 3 + 3χ 4 + 4χ 5 + 3χ 6 + 4χ 7 + 4χ {65} χ 1 + 2χ 2 + 2χ 3 + χ 4 + 2χ 5 + 3χ 6 + 2χ 7 + 2χ {641} 5χ 2 + 5χ χ χ χ χ χ {632} 2χ 1 + 8χ 2 + 8χ χ χ χ χ χ {631 2 } χ 1 + 8χ 2 + 8χ χ χ χ χ χ {62 2 1} 3χ 1 + 9χ 2 + 9χ χ χ χ χ χ {621 3 } χ 1 + 8χ 2 + 8χ χ χ χ χ χ {61 5 } 2χ 1 + 3χ 2 + 3χ 3 + 2χ 4 + 6χ 5 + 4χ 6 + 4χ 7 + 4χ {5 2 1} 3χ 2 + 3χ 3 + 4χ 4 + 5χ 5 + 6χ 6 + 6χ 7 + 6χ {542} 2χ 1 + 8χ 2 + 8χ χ χ χ χ χ {541 2 } 2χ 1 + 8χ 2 + 8χ χ χ χ χ χ {53 2 } χ 1 + 4χ 2 + 4χ χ χ χ χ χ {5321} 3χ χ χ χ χ χ χ χ {531 3 } 2χ χ χ χ χ χ χ χ {52 3 } 3χ 1 + 6χ 2 + 6χ χ χ χ χ χ {4 2 3} 2χ 1 + 3χ 2 + 3χ 3 + 8χ 4 + 7χ 5 + 8χ 6 + 8χ 7 + 8χ {4 2 21} 2χ χ χ χ χ χ χ χ {43 2 1} 8χ 2 + 8χ χ χ χ χ χ 8 Under S 11 A 11 the above irreducible representations are irreducible apart from the two irreducible representations {61 5 } and {43 2 1} which split as {61 5 } [61 5 ] + + [61 5 ] {43 2 1} [43 2 1] + + [43 2 1]

7 Notes on PSL(2, 11) I 7 Under A 11 PSL(2, 11) the branching rules are the same as in the above Table apart from the splitting cases where [61 5 ] + χ 1 + χ 2 + 2χ 3 + χ 4 + 3χ 5 + 2χ 6 + 2χ 7 + 2χ 8 [61 5 ] χ 1 + 2χ 2 + χ 3 + χ 4 + 3χ 5 + 2χ 6 + 2χ 7 + 2χ 8 [43 2 1] + 4χ 2 + 4χ χ 4 + 8χ χ χ χ 8 [43 2 1] 4χ 2 + 4χ χ 4 + 8χ χ χ χ Embeddings of PSL(2, 11) in the Orthogonal groups O n and SO n The group S n+1 may be embedded in the orthogonal group O n. In that case the n dimensional irreducible representation {n, 1} is embedded in the vector irreducible representation [1] of the full orthogonal group, O(n). The irreducible representation {n, 1} is orthogonal but not unimodular and hence S n+1 cannot be embedded in the special orthogonal group SO n. For n 5 all the irreducible representations of the alternating group are both orthogonal and unimodular and as such can be embedded in an appropriate SO n. For example, in the case of the icosahedral group I A 5 we may embed the irreducible representation U in the vector irreducible representation [10] of SO 4 or the irreducible representation V in the vector irreducible representation of SO 5 etc. For the group PSL(2, 11) the irreducible representations χ 2 + χ 3 and χ 5 are orthogonal and unimodular and hence may be embedded in SO 10. However, it is sufficient to note that SO 10 A 11 and one has the typical SO 10 A 11 decompositions Table 8 Some SO 10 A 11 branching rules Dim(λ) [λ] Decomposition 1 [0] [11 ] 10 [1] [10 1] 54 [2] [10 1] + [92] 45 [1 2 ] [91 2 ] 210 [3] [11 ] + [10 1] + [92] + [91 2 ] 320 [21] [92] + [91 2 ] + [821] 120 [1 3 ] [81 3 ] 660 [4] [11 ] + 2[10 1] + 2[92] + [91 2 ] + [83] + [821] + [74] 1386 [31] [10 1] + [92] + 2[91 2 ] + [83] + 2[821] + [81 3 ] + [731] 770 [2 2 ] [92] + [83] + [821] + [72 2 ] 945 [21 2 ] [821] + [81 3 ] + [721 2 ] 210 [1 4 ] [71 4 ] We note that in a similar fashion we have SO 4 A 5 and SO 7 A 8. In the latter case while A 8 L 168 one does not have G 2 A 8. In the former case it may be naturally seen from the fact that the pentahedral group P is a subgroup of SO(4) as noted elsewhere 4,5.

8 8 Brian G Wybourne 1.9. Spin irreducible representations of A n The alternating groups A n possess spin or projective irreducible representations which may be labelled by partitions of n into k distinct parts 1 3 with the understanding that irreducible representations with n k even constitute a pair of conjugate irreducible representations. Noting that SO n 1 A n we may equivalently label the projective irreducible representations of A n as [ ; λ] where λ is a partition into p distinct parts with the weight, ω λ, of λ is such that n ω λ > λ 1 and λ 1 < [ n ]. In that case 2 if the partition (λ) is such that n p is odd then the irreducible representation forms a conjugate pair and the two irreducible representations are distinguished by attaching a ± as a subscript to [ ; λ]. Thus for A 5 we have the projective irreducible representations and their dimensions as given below Table 9 Dimensions of the projective irreducible representations of A 5. Dim[ ; λ] Irreducible representation 2 [ ; 0] ± 6 [ ; 1] 4 [ ; 2] while for A 8 we have Table 10 Dimensions of the projective irreducible representations of A 8. Dim[ ; λ] Irreducible representation 8 [ ; 0] 24 [ ; 1] ± 56 [ ; 2] ± 64 [ ; 21] 56 [ ; 3] ± 48 [ ; 31] and finally for A 11 we have Table 11 Dimensions of the projective irreducible representations of A 11. Dim[ ; λ] Irreducible representation 16 [ ; 0] ± 144 [ ; 1] 560 [ ; 2] 616 [ ; 21] ± 1200 [ ; 3] 1584 [ ; 31] ± 1232 [ ; 32] ± 528 [ ; 321] 1440 [ ; 4] 1584 [ ; 41] ± 880 [ ; 42] ± 672 [ ; 5]

9 Notes on PSL(2, 11) I PSL(2, 11) as a subgroup of SU 5 The group PSL(2, 11) is a natural subgroup of the special unitary group SU 5 with the vector irreducible representation of SU 5, {1}, branching to the complex χ 2 irreducible representation of PSL(2, 11). The SU 5 PSL(2, 11) branching rules may be readily evaluated and a typical table is given below. Table 12 The SU 5 PSL(2, 11) branching rules. Dim(λ) SU 5 PSL(2, 11) 1 {0} χ 1 5 {1} χ 2 15 {2} χ 3 + χ 5 10 {1 2 } χ 4 35 {3} χ 1 + χ 5 + χ 7 + χ 8 40 {21} χ 3 + χ 6 + χ 7 + χ 8 10 {1 3 } χ 4 70 {4} 2χ 2 + χ 3 + 2χ 5 + χ 6 + χ 7 + χ {31} χ 2 + 2χ 4 + χ 5 + 2χ 6 + 2χ 7 + 2χ 8 50 {2 2 } χ 1 + χ 2 + 2χ 5 + χ 7 + χ 8 45 {21 2 } χ 4 + χ 6 + χ 7 + χ 8 5 {1 4 } χ {5} χ 1 + χ 2 + 2χ 3 + χ 4 + 3χ 5 + 2χ 6 + 2χ 7 + 2χ {41} χ 2 + 2χ 3 + 4χ 4 + 4χ 5 + 3χ 6 + 4χ 7 + 4χ {32} χ 2 + 2χ 3 + 2χ 4 + 3χ 5 + 3χ 6 + 3χ 7 + 3χ {31 2 } χ 2 + 3χ 4 + χ 5 + 3χ 6 + 2χ 7 + 2χ 8 75 {2 2 1} χ 2 + χ 3 + χ 4 + 2χ 5 + χ 6 + χ 7 + χ 8 24 {21 3 } χ 7 + χ 8 1 {1 5 } χ {6} 2χ 1 + 2χ 2 + 2χ 3 + χ 4 + 6χ 5 + 2χ 6 + 4χ 7 + 4χ {51} 3χ 2 + 3χ 3 + 7χ 4 + 4χ 5 + 8χ 6 + 8χ 7 + 8χ {42} 2χ 1 + 4χ 2 + 4χ 3 + 5χ 4 + 7χ 5 + 6χ 6 + 8χ 7 + 8χ {41 2 } χ 2 + 2χ 3 + 5χ 4 + 4χ 5 + 5χ 6 + 5χ 7 + 5χ {3 2 } 6χ 4 + χ 5 + 3χ 6 + 3χ 7 + 3χ {321} 2χ 2 + 3χ 3 + 3χ 4 + 5χ 5 + 5χ 6 + 5χ 7 + 5χ 8 70 {31 3 } χ 2 + 2χ 4 + χ 5 + χ 6 + χ 7 + χ 8 50 {2 3 } χ 1 + χ 3 + 2χ 5 + χ 7 + χ 8 45 { } χ 4 + χ 6 + χ 7 + χ 8 5 {21 4 } χ 2

10 10 Brian G Wybourne Table 12 The SU 5 PSL(2, 11) branching rules (continued). Dim(λ) SU 5 PSL(2, 11) 330 {7} χ 1 + 4χ 2 + 4χ 3 + 2χ 4 + 7χ 5 + 5χ 6 + 6χ 7 + 6χ {61} χ 1 + 6χ 2 + 5χ χ χ χ χ χ {52} 2χ 1 + 7χ 2 + 6χ χ χ χ χ χ {51 2 } 3χ 2 + 3χ χ 4 + 6χ χ χ χ {43} 5χ 2 + 5χ 3 + 9χ 4 + 7χ χ χ χ {421} 2χ 1 + 6χ 2 + 5χ χ χ χ χ χ {41 3 } χ 2 + 2χ 3 + 2χ 4 + 2χ 5 + 3χ 6 + 3χ 7 + 3χ {3 2 1} χ 2 + 2χ 3 + 6χ 4 + 3χ 5 + 6χ 6 + 6χ 7 + 6χ {32 2 } χ 1 + 2χ 2 + 2χ 3 + 2χ 4 + 4χ 5 + 3χ 6 + 4χ 7 + 4χ {321 2 } χ 2 + χ 3 + 3χ 4 + 3χ 5 + 3χ 6 + 3χ 7 + 3χ 8 15 {31 4 } χ 3 + χ 5 40 {2 3 1} χ 2 + χ 6 + χ 7 + χ 8 10 { } χ SO(10) P SL(2, 11) Decompositions Table 13 Some SO(10) P SL(2, 11) Decompositions Dim[λ] SO(10) P SL(2, 11) 1 [0] χ 1 10 [1] χ 2 + χ 3 45 [1 2 ] χ 1 + 2χ 4 + χ 7 + χ 8 54 [2] χ 2 + χ 3 + 2χ 5 + χ 7 + χ [3] χ 1 + χ 2 + χ 3 + 4χ 4 + 4χ 5 + 2χ 6 + 4χ 7 + 4χ [21] 3χ 2 + 3χ 3 + 6χ 4 + 2χ 5 + 6χ 6 + 6χ 7 + 6χ [1 3 ] χ 2 + χ 3 + 4χ 4 + 2χ 6 + 2χ 7 + 2χ [4] χ 1 + 8χ 2 + 8χ 3 + 5χ χ χ χ χ [31] χ χ χ χ χ χ χ χ [2 2 ] 4χ 1 + 6χ 2 + 6χ χ χ χ χ χ [21 2 ] 6χ 2 + 6χ χ χ χ χ χ [1 4 ] χ 1 + 3χ 2 + 3χ 4 + 2χ 5 + 3χ 6 + 4χ 7 + 3χ 8 References [1] Frobenius G (1901) S. B. Preuss Akad. Wiss. 303 [2] Dehuai Luan and Wybourne B G (1981) J. Phys. A: Math. Gen [3] Dehuai Luan and Wybourne B G (1981) J. Phys. A: Math. Gen [4] Backhouse N B and Gard P (1974) J. Phys. A: Math. Gen [5] Bickerstaff R P and Wybourne B G (1978) J. Phys. A: Math. Gen