EJcr'ENDED TABLES OF ZONAL POLlNOMIALS
by
James O. Kitchen
university of North Carolina
Institute of Statistics Mimeo Series No. 565
February, 1968
''--
This research was supported by the National
Science Foundation Grant No. GP-5790, and the Air
Force Office of Scientific Research Grant NO.
AFOSR-68...l4l5.
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
1.
Introduction
Zonal polynomials are defined by James [2] in a manner similar
to the following:
Let V be the vector space of hOJOOgeneous polynomials ~(s) of
k
degree k in the jm(m + 1) different elements of a s;ymm.etrix m X m
matrix S.
L
E
Let G(m) be the group of non-singular m X m matrices.
For
G(m), define an induced linear trMsformation T of the polyL
nomials ~(s) by
T : '(8) -+(Lt)(8) = ~(L-l 8L-l ')
L
A representation of G(m) is then given by the homomorphism T where
A for~ definition of a
Md is given by T(L) = T :
L
T:G(m)-+ Vk
representation may be found on 'page 312 of Hew!tt and Ross [1].
A subspace V' c V is called invariant if and only if LV'cV'
k
for all L
E
G(m).
V' is an irreducible invariMt subspace if and only
if it has no proper invariMt subspace.
Let P(k,m) be the set of partitions p
into q non-zero parts k
Md where q
= kl
+ k
2
= (~,k2, ••• ,kq)
+ ••• + k q where k
i
~
of k
k j for i < j
S m.
Using theorem 3 of Thrall [3], James concludes that Vk CM be
decomposed into a direct sum of irreducible invariMt subspaces
Vk,p' one corresponding to each element of P(k,m), i.e.
v =
k
(i)
peP(k,m)
V
k,p
The zonal polynomial C (8) is then defined to be the component of
k ,p
(tr S)k 'Which belongs to the subspace Vk,p' so that
(tr 8)k
=)'
p~k,m)
Ck,p(S).
....
2
Let us now define
and
-1
(1.1)
J
Then, the K,p
z. (S) differ from Ck ,p (S) only by a multiplicative constant and are also called zonal polynomials.
this paper, the
~,p'
As is the case in
which have leading coefficient unity, are the
polynomials which are usually tabulated.
They are usually written
as functions of
(1.2)
m
where {tili=l are the eigenroots of S.
2.
Tables.
The following tables give the zonal polynomials ~,p(S) defined
in (1.1) for k = 1 through k = 9.
The polynomials for each k are
given in the table identified as "Table k."
Each table presents the
polynomial corresponding to the partition of k, listed at the left
of the table, by means of coefficients of terms in s j ' listed at the
top of the table, where s j is defined by (1.2).
The tables for k = 1 through k = 6 are taken from James [2]
and are included here for the sake of completeness.
k = 7 through k
= 9 were
The tables for
obtained using a computer program written
by the author and based on the numerical technique outlined by James
[2].
In addition, a procedure was included within the program to
3
check the results for one of the 'cr'thogona1ity' conditions given by
James.
The program was tested by generating the
po~omia1s
for
k = 2 through k = 6 and comparing the results with those of James.
The polynomials for k = 7 through k =
9 were then generated.
TABLES OF ZONAL POLYNOMIAI.S
2
Table
Table 1
I~
~
Table 3
2
.2
&1
1
,s2
2
2
1
1
~~
Z
z
-'
21
3
1
s3
1
1
sls 2
6
~
1
1
-2
1
-3
2
Table 4
2
1 2
s2
2
1
12
1
31
22 '1
2
21
1
4
1
1
5
2
s4
1
Z
4
8 S
8
8.
84
12
1 3
32
-2
4
-8
-8
-2
-1
7
-2
-2
4
-6
3
8
-6
48
Table 5
S?3
8/ 3
8 8
1 4
20
2
1 2
.. 60
80
160
240
384
1
11
6
g6
-20
24
-48
32
2
31
221
1
6
11
-4
20
-26
-8
1
-10
2
-4
-8
16
1
3
0
5
-10
-10
10
4
213
15
1
-4
-3
2
10
6
-12
1
-10
15
20
-20
-30
24
z
5
41
\----
8
5
1
s3s
12
1
8 8
8
S
s3
4
Table 6
Z
6
51
42
412
32
321
313
23
2212
214
16
S6
'1
1
1
1
1
1
1
1
1
1
1
1
,4,
8 8
1 2
30
19
12
9
9
4
0
0
-3
-8
-15
S2s 2
1 2
180
48
27
-12
33
3
-21
15
3
3
45
83 8 3S
2 1 3
120 160
-12 72
30 16
-12 22
-27 -8
-2
-8
6
4
30 -20
: -9
-8
6 12
-15 40
8 8 8
1 2 3
960
80
24
-60
120
0
12
-60
0
20
-120
2 28
3 s1 4 8 284 sls 5 s6
640 720 1440 2304 3840
-64 192 -144 192 -384
-8 -18 108 -144 -48
16
12
-24 -48
96
136 -78 -1l4 -48
-24
-4 32
16
-24 -18
16
-6
-48
12 24
40
0
30 -60 24
4
24 -24
24
-12
48
-16
-6 -36 -24
-40 -90
-120
90 144
8
Table 7{a)
Z
7
61
52
51 2
43
421
413
321
2
32
2
321
4
31
231
2
2 13
215
17
5.7
8 58
1 2
8~82
1 2
8
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
42
29
20
17
15
10
6
7
4
1
-4
-3
-7
420
160
79
16
69
9
-39
21
15
840
60
114
-84
39
14
-18
-49
50
-13
42
15
-21
15
-105
1
-13
-21
-9
-29
15
7
25
105
83
1 2
,4,
S18 3
280
150
60
66
10
10
22
-14
-20
-8
12
-20
0
30
70
,2.
~'lS2S3
3360
760
148
-56
228
-42
-78
84
-60
12
52
-60
28
-20
-420
,2,
82
1 3
S2$'3
8
3360
-280
368
-136
-42
-52
84
14
80
-16
-16
4480
320
-184
-64
376
-64
112
56
-40
-28
32
100
-28
-40
280
-60
56
-70
210
5
L
Table 7(b)
z
7
61
52
2
51
43
421
413
321
2
32
3212
4
31
3
2 1
2213
215
. 7
1
8 3S
14
1680
640
118
160
-102
-22
6
-70
-10
-4
-14
60
28
-50
-210
12 4
8:;S4
s28
1'5
10080
720
180
-432
13440
-1120
-112
224
588
-112
48
-196
80
68
8064
1824
-120
96
-360
-100
-36
8 8 8
90
100
-36
-182
-140
64
-36
0
0
-90
630
56
-60
104
44
24
-,6
-28
140
-420
-84
24
504
-112
8 8
8 8
16128
-1344
816
-19'2
-504
-24
72
168
-144
12
-48
108
-84
168
-504
26880
1920
-1104
-384
-264
176
144
2 5
1 6
56
80
-76
-96
-60
84
120
-840
s7
46080
-3840
-384
768
-144
96
-288
48
0
-48
192
0
48
-240
720
6
Table 8(a)
Z
8
71
62
61 2
53
521
513
42
431
422
4212
4
41
2
3 2
2 2
3 1
2
132 1
3213
315
24
2312
22 14
216
~el
sel
1
6,
sls 2
s4s 2
12
s2s 3
1 2
s4
2
5
sls 3
,3,
sls t!3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
56
41
30
27
23
18
14
20
13
10
840
390
203
110
147
47
-33
138
47
23
-19
-69
35
2
5
-19
-25
30
21
26
75
210
3360
660
396
-180
291
36
-180
156
-19
116
1680
-120
276
-120
-102
-12
36
321
-22
116
-34
36
-70
41
..10
6
-30
165
-42
33
-30
105
448
268
136
142
52
52
64
16
2
-4
8
28
-28
-16
-22
-2
28
-32
-14
16
58
112
8960
2960
848
440
512
-28
-184
608
132
-132
-72
-52
0
96
-60
68
80
-160
-28
32
-220
-1120
7
2
7
4
1
-3
-9
-4
-7
-12
-19
-28
-19
36
35
-100
35
-9
135
60
-21
-36
-15
-420
2
sls~3
26830
1680
1504
-11~00
1140
40
-48
-336
-210
24
-48
312
420
-48
-30
22
-260
-480
42
112
-210
1680
7
I
"-
Table 8(b)
Z
8
71
62
612
53
521
513
42
431
42 2
4212
414
322
3212
13221
3213
315
24
2312
2214
216
18
22
13
17920
3520
-176
160
496
-344
160
1504
216
-216
-36
304
0
0
0
-56
40
400
8 8
112
-128
40
1120
82
23
35840
-2560
1312
-320
1256
-464
512
-1888
16
112
64
-256
280
-128
-140
124
-80
560
-28
-128
320
-1120
8
,4,
8 8
14
3360
1560
526
580
-6
94
138
-204
-106
-34
-16
-6
-70
-52
20
4
-50
120
84
4
-150
-420
2
1 2$4
40320
7920
1320
-480
522
-48
-792
360
-18
0
216
-144
-630
-72
0
96
-210
0
0
-120
8 8
90
2520
2
24
40320
-2880
3192
-1200
-1260
-120
360
2412
-80
232
-32
-72
-140
100
-20
-12
60
-720
252
-228
360
-1260
8 S
S183~4
107520
6720
-3136
-1120
2184
-256
960
4704
-196
-256
8
-192
-280
-160
320
8
-280
-480
-336
224
420
-3360
2
4
80640
-5760
-480
960
-144
96
-288
5580
-664
-160
272
-288
560
92
-160
-48
240
540
0
60
-360
1260
8
3
15
21504
7104
1120
1504
-672
-152
48
-960
-232
-40
8 8
-64
-24
224
128
104
16
64
-96
-168
-128
264
1344
8
Table
Z
8
71
62
612
53
521
513
42
431
422
4212
414
322
22
31
13221
3213
315
24
~12
2214
216
1°
s18 2s 5
129024
8064
1728
-4032
720
720
-288
-4032
-280
-592
104
144
-112
512
-16
-216
144
864
0
0
504
-4032
5 5
3 5
172032
-12288
-1024
2048
4128
-832
384
-4224
-192
576
48
-192
-672
384
72
-232
512
-768
168
128
-768
2688
8Cc)
2
1 6
107520
21120
-1056
960
-2568
-768
-288
-1056
232
544
244
144
280
-128
-140
-156
-120
-240
84
384
-120
-3360
8 5
5 2s 6
215040
-15360
7872
-1920
-3552
-192
576
-1248
768
-768
72
-288
0
-384
240
-48
240
240
-336
384
-960
3360
s15 7
368640
23040
-10752
-3840
-2016
1344
1152
-1152
304
448
-416
-576
160
-128
-200
264
480
0
216
-384
-720
5760
s8
645120
-46080
-3840
7680
-1152
768
-2304
-720
288
0
-288
1152
0
-144
0
192
-960
0
0
-240
1440
-5040
9
I
"-
Table 9(a)
z
9
81
72
712
63
621
613
54
531
2
52
5212
4
51
421
432
4312
2
42 1
4213
415
33
2
3 21
3213
323
32212
3214
6
31
241
2313
115
217
19
s9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
8 78
12
72
55
42
39
33
28
24
28
21
18
15
10
18
13
10
7
3
-3
9
4
0
0
-3
-8
-15
-8
-12
-18
-26
-36
s5 s2
12
1512
798
447
318
303
153
33
258
III
63
-3
-93
108
63
12
-3
-51
-93
63
18
-18
18
-3
-18
3
42
42
72
168
378
s3 s3
1 2
10080
2940
1380
60
993
148
-540
708
141
228
-99
-204
18
133
-134
85
-15
225
105
-20
-144
120
15
40
285
0
-84
-90
-2+0
-1260
.4
8 S
12
15120
840
1620
-loBo
378
108
-180
633
-270
468
-126
324
243
98
-61
50
-54
90
-630
-55
189
225
-90
45
-270
105
-63
135
-105
945
.6
sl s3
672
434
252
258
126
126
138
56
42
36
48
68
6
-14
-2
-8
12
42
-42
-32
-12
-36
-18
12
54
-28
0
42
98
168
10
Table 9(b)
Z
9
81
72
2
71
63
621
613
54
531
522
521
4
51
421
432
2
431
4221
4213
415
33
3221
2
3 13
323
32212
4
321
6
31
241
~J?
2215
21 7
19
,4,
8 8 8
12 3
20160
8260
3060
2340
1440
470
90
1240
288
-180
-180
-260
450
0
90
-180
-36
0
0
0
180
-180
0
140
0
-140
0
-90
-770
-2520
22
S18 2S3
120960
21000
6960
-3000
5754
-96
-2400
2424
366
144
-96
624
-1026
294
-426
-120
204
390
1890
240
-36
-540
30
-60
-1110
-420
336
90
210
7560
3
23
80640
-5040
7440
-3120
8 8
-948
-568
1080
2232
-204
1392
-360
240
-558
-28
170
-280
240
-540
-420
80
-180
420
-60
20
300
-420
336
-450
630
-2520
3 2
1 3
53760
15680
2160
2880
720
-640
480
2720
144
-840
-276
704
1200
0
144
-180
60
600
0
0
-96
240
0
-160
120
560
0
-240
560
3360
8 8
2
s18 28 3
322560
17920
3360
-7680
11496
-1344
1920
-224
168
-1584
-96
1024
-2304
1176
-96
0
240
-1680
2520
-480
144
0
-60
240
240
1120
-672
0
1120
-10080
s3
3
143360
-8960
-640
1280
6704
-1536
1280
-3776
-30!~
704
128
-512
944
-336
48
0
-64
320
3920
-480
368
560
80
-240
320
-560
224
80
-560
2240
11
Table 9(c)
z
9
81
72
2
71
63
621
613
54
531
522
5212
4
51
421
432
4312
4221
4213
415
33
2
3 21
3213
323
32212
4
321
31°
241
2313
2215
217
19
5
1$4
6048
3192
1398
1464
354
474
534
-156
-30
54
8
84
114
-216
-126
-96
-12
-12
-42
-126
-36
-36
72
48
-12
-138
168
84
-72
-336
-756
P
8,3,
18 2 4
120960
35280
8760
4560
2478
248
-2040
1368
150
-72
-168
-1368
108
-602
172
280
312
-510
-1890
-440
72
0
120
-40
-390
0
-168
-180
1260
7560
.,
88 8
1 24
362880
20160
15480
-14400
3924
1704
-1800
2124
-2412
504
144
1224
2484
84
-36
240
-288
-180
-1260
-60
324
-1440
360
-420
1260
0
252
-540
1260
-11340
28 8
134
483840
84000
-3360
3360
708
-4032
960
14448
588
-1152
492
1632
2268
-1092
-468
300
-132
-1020
-1260
240
-288
720
240
0
-420
-1680
-336
1260
-420
-15120
8
828'38 4
967680
-60480
26880
-6720
5784
-4256
5280
3024
-1176
-576
1464
-3456
-756
2184
-516
-840
408
600
-7560
240
144
0
240
-560
840
0
-336
1260
-3780
15120
8 8
2
1 4
725760
40320
-15840
-5760
-2448
1632
1440
19692
-2736
-288
1008
-2592
2052
-1008
-492
240
432
-720
5040
540
108
-180
-720
300
720
1260
252
-540
-1260
11340
Table 9(d)
z
9
81
72
712
63
621
613
54
531
522
5212
4
51
421
432
4312
2
42 1
4213
415
33
2
3 21
3213
323
32212
3214
6
31
4
21
2313
2215
217
19
,4,
sls5
48384
19824
5784
6384
24
884
1284
-1776
-600
-216
-156
24
-756
-56
-116
4
-60
24
504
304
144
144
24
-16
264
-336
-336
-36
924
3024
2
sl s2s5
580608
100800
14688
-5184
4968
-432
-7344
-5472
1080
288
972
-288
-3672
-1512
552
-372
-36
648
-504
576
432
720
-432
-432
1080
1008
0
648
-504
-18144
2
s2 s5
580608
-36288
34848
-13248
-10944
-1104
3312
5616
576
2016
-792
288
-1404
-1344
708
168
-216
288
4032
-48
-432
-1008
288
48
-288
1008
-1008
1188
-2268
9072
s
13 5
1548288
86016
-33792
-12288
19488
-2432
8832
6528
1824
-1152
-1392
768
-6912
-672
768
1488
-912
768
-6048
192
288
-864
-312
-32
1536
-672
1344
-1152
-2688
24192
8 S
3
sl s6
s4 8 5
2322432 322560
94080
-145152
-10368
12960
17280
20736
-2592
-5976
1728
-1376
480
-5184
31968
-5856
-4320
-1656
-1152
-336
1728
-492
-1152
-192
144
-7992
1064
1568
1928
368
-448
500
228
-1152
120
2592
2016
840
-1024
-160
-432
-432
1008
-480
432
-300
432
-80
-360
-2592
0
-1008
0
672
-648·
720
4536
-1680
-18144 -10080
13
Table 9(e)
z
9
81
72.
2
71
63
621
61'
54
531
522
5212
4
51
4~
432
4312
4221
4213
415
33
3221
3213
3~
~2212
3214
31°
241
~13
2215
217
19
1 286
1935360
107520
20160
-46080
7200
6720
-2880
-23520
-2016
-4320
792
1152
4320
0
1152
-360
-360
-720
0
0
-1728
2160
0
960
-720
-1680
0
8 8
0
-3360
30240
386
2580480
-161280
-11520
23040
36304
-7936
3840
-23616
-1440
4224
384
-1536
5904
-2016
144
0
-192
960
-3360
1440
-1296
-1680
-240
1040
-2880
1680
-672
-720
5040
-20160
8
2
1 7
1658880
288000
-11520
11520
-24048
-7488
-2880
-7488
1584
4032
1872
1152
432
1392
-528
-720
-864
-720
720
-480
432
-720
360
720
720
720
-288
-2160
720
25920
8 8
88
27
3317760
-207360
92160
-23040
-33120
-1920
5760
-8640
5472
-5760
576
-2304
2160
-160
-1744
1280
-288
1440
1440
-160
1296
720
-720
240
-1440
-720
1440
-2160
6480
-25920
8188
5806080
322560
-126720
-46080
-19584
13056
11520
-8784
2304
3456
-3168
-4608
1296
896
-688
-1120
1440
2880
0
-400
432
0
720
-1200
-2880
0
-1008
2160
5040
-45360
8
9
10321920
-645120
-46080
92160
-1l520
7680
-23040
-5760
2304
0
-2304
Cl216
1440
0
-864
0
1152
-5760
0
0
576
0
0
-960
5760
0
0
1440
-10080
40320
14
REFERENCES
(lJ
He"Titt, E. and Ross, K. A. (1963).
Abstract Harnxmic Analysis.
Academic Press, Ne'" York.
(2J
James, A. T. (1964).
Distributions of matrix variates and
latent roots derived from normal samples.
Ann. Math.
Statist., 35, 475-501.
(3 J
Thrall, R. M. (1942).
On synIIlEtrized Kronecker po't-Ters and
the structure of the free Lie ring.
64, 371-388.
Amer. J. Math.,