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Power sum of character degrees of abelian wreath products

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Numbers
$q$
$n$
$s$ 
$D_{q,n}(s)$
1
3
0:
3
1
3
1:
4
1
3
2:
6
1
3
9/4:
6.756828460010884266869999882241903661171888369855269652076008898877866672907668639483125378152550695
1
3
3:
10
1
3
4:
18
1
3
-2:
9/4
1
3
-4:
33/16
1
4
0:
5
1
4
1:
10
1
4
2:
24
1
4
9/4:
30.44616069315574856161594011458788665405312299356006883325061675360346898645910703439650208259138546
1
4
3:
64
1
4
4:
180
1
4
-2:
89/36
1
4
-4:
2705/1296
1
5
0:
7
1
5
1:
26
1
5
2:
120
1
5
9/4:
178.3653179396384120542904668514688414355886578326900676389965539304891565373410282003568921726354748
1
5
3:
596
1
5
4:
3060
1
5
-2:
4019/1800
1
5
-4:
13036361/6480000
1
6
0:
11
1
6
1:
76
1
6
2:
720
1
6
9/4:
1299.782990956064735987720445776579631928238532159203367899066225231265671744424068272068204681163548
1
6
3:
8056
1
6
4:
101160
1
6
-2:
1144937/518400
1
6
-4:
539336815121/268738560000
1
7
0:
15
1
7
1:
232
1
7
2:
5040
1
7
9/4:
11209.92455592916320834832138041207575602769783104826665130323816269542442381331702418737742411822057
1
7
3:
130432
1
7
4:
3807720
1
7
-2:
41033/19600
1
7
-4:
593209193/296352000
1
8
0:
22
1
8
1:
764
1
8
2:
40320
1
8
9/4:
112179.3457758479302571074093230299018041481952260379430517870390357996217670517196438259206549899894
1
8
3:
2534960
1
8
4:
174986280
1
8
-2:
420033497/203212800
1
8
-4:
165257187092529521/82590884167680000
1
9
0:
30
1
9
1:
2620
1
9
2:
362880
1
9
9/4:
1285807.039957291442948297487883554439354097952463452976076127792822348684204936652527004314334156908
1
9
3:
59822884
1
9
4:
10699554600
1
9
-2:
419901617/205752960
1
9
-4:
846895994783386013/423342805487616000
1
10
0:
42
1
10
1:
9496
1
10
2:
3628800
1
10
9/4:
16667921.44979719062065717637343047052634312158067959050491217807688275543452610661371064563480762644
1
10
3:
1718480368
1
10
4:
927701102160
1
10
-2:
5347128272053/2633637888000
1
10
-4:
346855859144796627869129777/173401213127727513600000000
1
11
0:
56
1
11
1:
35696
1
11
2:
39916800
1
11
9/4:
240658052.5424246495531230612198817645824648200521432951770237722553991578393318348250691254558574699
1
11
3:
56754444440
1
11
4:
95030461809360
1
11
-2:
77711622041/38419920000
1
11
-4:
159433773272852104160251/79708873651545600000000
1
12
0:
77
1
12
1:
140152
1
12
2:
479001600
1
12
9/4:
3829972410.229332366404897478065877281346072481362905388348773917921242013813387446321235363260836513
1
12
3:
2110577206816
1
12
4:
10905467528783520
1
12
-2:
8269250393826847/4097188085760000
1
12
-4:
3290467738562597982893555964761041/1645121120589181925248204800000000
1
13
0:
101
1
13
1:
568504
1
13
2:
6227020800
1
13
9/4:
66689724267.86330420965388938765456972084202247948904789149304338320745183796236233234369984972283481
1
13
3:
87981286785328
1
13
4:
1431935974242053280
1
13
-2:
235496224142051/116873396640000
1
13
-4:
737642757799273130826419688217/368803552738610608819200000000
1
14
0:
135
1
14
1:
2390480
1
14
2:
87178291200
1
14
9/4:
1263847451275.143000161055659959468520861100504932330220983005784382202853104914467460604980423881948
1
14
3:
4129351961475872
1
14
4:
222906109589537774400
1
14
-2:
955985490112174877777/475003403534499840000
1
14
-4:
90254458336480038242313913502316465783757/45125646673871779018320223792005120000000
1
15
0:
176
1
15
1:
10349536
1
15
2:
1307674368000
1
15
9/4:
25930944682565.26033672610840582995731678317226408334615795802705240934043404401219290706942428621509
1
15
3:
218382856010529472
1
15
4:
42471495822490670295360
1
15
-2:
8489838027231281719093/4222252475862220800000
1
15
-4:
18082373226427714863958182093104477214893897/9040950999022645433172095926272000000000000
1
16
0:
231
1
16
1:
46206736
1
16
2:
20922789888000
1
16
9/4:
572749613038644.0698459852756585655391755828370136958445043773214778747456209958641338198492198057928
1
16
3:
12813477368159567200
1
16
4:
9447237366839585591438160
1
16
-2:
2290598610488721737306051/1140008168482799616000000
1
16
-4:
202541971567170592285870633095825788949312483943/101268983704481083303451099626733568000000000000
1
17
0:
297
1
17
1:
211799312
1
17
2:
355687428096000
1
17
9/4:
13548530678677162.20668805069955304793019881116182992834550295381250347624817850214743156492279117466
1
17
3:
822337333595479929044
1
17
4:
2329156499421828313498781520
1
17
-2:
31756632017799008765674036369/15814193313193396273152000000
1
17
-4:
1000370114626898126746395232292588909023117302741109631521/500177420294101456136632723567242165992030208000000000000
1
18
0:
385
1
18
1:
997313824
1
18
2:
6402373705728000
1
18
9/4:
341774695045882406.4458787104496575671347364313061396919600535841482753677365030002126888970013541364
1
18
3:
57213666993723455063392
1
18
4:
625767950674327348244399122080
1
18
-2:
6679598050200775906215769/3327902633247768576000000
1
18
-4:
86837397901004863317080651438224862992328280996058981/43418178844974084734082701698545326909030400000000000
1
19
0:
490
1
19
1:
4809701440
1
19
2:
121645100408832000
1
19
9/4:
9163582824134146732.998082491030676654807598085260177159398280278068148518794472898547964220270578988
1
19
3:
4305630141314873304140008
1
19
4:
184424930226363787003768804279200
1
19
-2:
1527215066227242164483950050527/761189838141708807281049600000
1
19
-4:
469326547942973727886753654806171233624453506782609423063522037/234661037724531345114655607404823638731290527452364800000000000
1
20
0:
627
1
20
1:
23758664096
1
20
2:
2432902008176640000
1
20
9/4:
260425955514691667230.3969666034967925479078839224057776360850522856661073472590126419411205575485766
1
20
3:
350765912068457786334084992
1
20
4:
60597106700643736394765182697908800
1
20
-2:
45286798239201554820617937026669/22579239583549223653478400000000
1
20
-4:
522061796789901029432346828768669618876223910515091752262503437709/261028894807713852826818752743707935539414664478720000000000000000
2
2
0:
5
2
2
1:
6
2
2
2:
8
2
2
9/4:
8.756828460010884266869999882241903661171888369855269652076008898877866672907668639483125378152550695
2
2
3:
12
2
2
4:
20
2
2
-2:
17/4
2
2
-4:
65/16
2
3
0:
10
2
3
1:
20
2
3
2:
48
2
3
9/4:
60.89232138631149712323188022917577330810624598712013766650123350720693797291821406879300416518277092
2
3
3:
128
2
3
4:
360
2
3
-2:
89/18
2
3
-4:
2705/648
2
4
0:
20
2
4
1:
76
2
4
2:
384
2
4
9/4:
592.0436512136938768506739923731507960795586969341512972040471239977531293743070472065378305922768038
2
4
3:
2272
2
4
4:
14760
2
4
-2:
1537/288
2
4
-4:
695665/165888
2
5
0:
36
2
5
1:
312
2
5
2:
3840
2
5
9/4:
7439.323671189014238722614912254623305478554771551795079844110636515394594340088448748998190344777496
2
5
3:
57192
2
5
4:
951120
2
5
-2:
713/150
2
5
-4:
725593/180000
2
6
0:
65
2
6
1:
1384
2
6
2:
46080
2
6
9/4:
115244.0133492895867631225620429086048961443203793406411261036812149545203324207119687045420683851664
2
6
3:
1937584
2
6
4:
96423840
2
6
-2:
2383523/518400
2
6
-4:
215911884199/53747712000
2
7
0:
110
2
7
1:
6512
2
7
2:
645120
2
7
9/4:
2129061.589061843474780365123211573185939511448786676769617805302514461290101310820531362429963760177
2
7
3:
82745504
2
7
4:
12597883200
2
7
-2:
10939253/2540160
2
7
-4:
258427409756573/64524128256000
2
8
0:
185
2
8
1:
32400
2
8
2:
10321920
2
8
9/4:
45830009.93510977715663754535439838667254277707682607045325282243443977052536599913418396532126979943
2
8
3:
4344284256
2
8
4:
2141549298000
2
8
-2:
85655611/20321280
2
8
-4:
33059626144332583/8259088416768000
2
9
0:
300
2
9
1:
168992
2
9
2:
185794560
2
9
9/4:
1129307384.080030733082302238544433756341564316063808574254698841954503451699880194153420953824519936
2
9
3:
276258511784
2
9
4:
486867788618400
2
9
-2:
54532251469/13168189440
2
9
-4:
1387767649499182034633/346802426255455027200
2
10
0:
481
2
10
1:
921184
2
10
2:
3715891200
2
10
9/4:
31400559788.10530485550479622607300767253627314823939214493260781478126402383143026255845075333102732
2
10
3:
20788976906464
2
10
4:
140096982057721920
2
10
-2:
9012571761667/2194698240000
2
10
-4:
38543413460110584919092457/9633400729318195200000000
2
11
0:
752
2
11
1:
5222208
2
11
2:
81749606400
2
11
9/4:
973776488583.2146268011358483205567976218592214933985065832584402080933465006281197164723925178393248
2
11
3:
1819164435706752
2
11
4:
49549674242587559040
2
11
-2:
464589521029967/113810780160000
2
11
-4:
5078392739049371405648371200881/1269383580701529263308800000000
2
12
0:
1165
2
12
1:
30710464
2
12
2:
1961990553600
2
12
9/4:
33357417441844.49287745391199475331123798939188751643920295853097196119282548682246410001608227752539
2
12
3:
182708270861062720
2
12
4:
21387067506678457584000
2
12
-2:
933087363080017457/229442532802560000
2
12
-4:
42120022750087231742084412734156269/10528775171770764321588510720000000
3
2
0:
9
3
2
1:
12
3
2
2:
18
3
2
9/4:
20.27048538003265280060999964672571098351566510956580895622802669663360001872300591844937613445765208
3
2
3:
30
3
2
4:
54
3
2
-2:
27/4
3
2
-4:
99/16
3
3
0:
22
3
3
1:
54
3
3
2:
162
3
3
9/4:
218.7495236615401819661271190105144420939892427916870160647888011373768577779353886026429061498671814
3
3
3:
570
3
3
4:
2322
3
3
-2:
73/9
3
3
-4:
2053/324
3
4
0:
51
3
4
1:
270
3
4
2:
1944
3
4
9/4:
3292.877668090042491837600151562717231910597623792756989274780932876897760548543627805550460388341176
3
4
3:
16992
3
4
4:
168156
3
4
-2:
829/96
3
4
-4:
349441/55296
3
5
0:
108
3
5
1:
1458
3
5
2:
29160
3
5
9/4:
64205.36909788274730898242224429179913929432004954558684017035965995259548106627337181880990402659411
3
5
3:
733788
3
5
4:
21204180
3
5
-2:
12147/1600
3
5
-4:
46524809/7680000
3
6
0:
221
3
6
1:
8424
3
6
2:
524880
3
6
9/4:
1546198.423057819690033235190097179552101858208551360454352306488233185602203998077929486317221195779
3
6
3:
42724584
3
6
4:
4050608040
3
6
-2:
372443/51840
3
6
-4:
810335220523/134369280000
3
7
0:
429
3
7
1:
51516
3
7
2:
11022480
3
7
9/4:
44496527.66842120941033577142254574296958946772800713613214063306664846709959260386974724868139221323
3
7
3:
3203890560
3
7
4:
1122379425720
3
7
-2:
56280443/8467200
3
7
-4:
1292692057195043/215080427520000
3
8
0:
810
3
8
1:
331452
3
8
2:
264539520
3
8
9/4:
1493593014.459235980374692813333577861978101927505775864437626203736760143542310938215089969135548724
3
8
3:
297746551440
3
8
4:
416543996163960
3
8
-2:
25883071/4014080
3
8
-4:
39191488517781479/6525699489792000
3
9
0:
1479
3
9
1:
2230740
3
9
2:
7142567040
3
9
9/4:
57409946783.83052894230125139347927678189183120322782714272889862092297302861084146471552691016557125
3
9
3:
33367171906956
3
9
4:
199746918188191080
3
9
-2:
92222764783/14631321600
3
9
-4:
3855502650888152109877/642226715287879680000
3
10
0:
2640
3
10
1:
15641424
3
10
2:
214277011200
3
10
9/4:
2490708146853.219192038617200590727298085346089691785336269838386761420832589985749689076402956013777
3
10
3:
4413783880638288
3
10
4:
118287889019146259280
3
10
-2:
27344417896723/4389396480000
3
10
-4:
69385117702438031273806967/11560080875181834240000000
8
2
2:
128
8
2
9/4:
149.1911968803047594723599967027733025128128743559475502581282491685802668414147219055275105882714195
8
2
3:
240
27
2
2:
1458
27
2
9/4:
1723.646789463820377671369958666908185071332817819199647878679123506131202190591692458577007731545294
27
2
3:
2862
64
2
2:
8192
64
2
9/4:
9717.766175381942682009919762599677780922526953628223618585233940137779212581859977197980762355542201
64
2
3:
16256
125
2
2:
31250
125
2
9/4:
37115.42056508435306824249908737475337408213486637833980358906896630346671503443195599422168068226789
125
2
3:
62250
216
2
2:
93312
216
2
9/4:
110885.5568414527326767213972656570030124112479480393613212049266319440641449160658087981712807022271
216
2
3:
186192
343
2
2:
235298
343
2
9/4:
279688.2596650183949047261030931343754387147685571211309032141499458835139660534887116037528047815559
343
2
3:
469910
512
2
2:
524288
512
2
9/4:
623293.2718247838362548659045953568693398617489909869548059751801156070066830895807426245294684040717
512
2
3:
1047552
729
2
2:
1062882
729
2
9/4:
1263710.972834648205519555688752182587913927610271314933796281417370635188856087319498684217845048242
729
2
3:
2124306
1000
2
2:
2000000
1000
2
9/4:
2378035.815775436691301564941179830878755358240742707191211966444989494403117380485421821126387199072
1000
2
3:
3998000
1331
2
2:
3543122
1331
2
9/4:
4213002.222382533827870639945770542559058155974484446998097256616530282960190671127836106519083494908
1331
2
3:
7083582
1728
2
2:
5971968
1728
2
9/4:
7101252.936379120719356199184289847226137087449535403795412871006264833442912373791694676904252009163
1728
2
3:
11940480
2197
2
2:
9653618
2197
2
9/4:
11479319.83505501618227610193593143765326691334593208611332086872281647104225520650503698024846966696
2197
2
3:
19302842
2744
2
2:
15059072
2744
2
9/4:
17907317.19909112180640149007682965127083964000355984238754424358600136792535402852715623611563779667
2744
2
3:
30112656
3375
2
2:
22781250
3375
2
9/4:
27090347.44062947093395770307952955873283979291981720967280126616687349363553392485747710973118381645
3375
2
3:
45555750
4096
2
2:
33554432
4096
2
9/4:
39901619.28958888155716126621241465956863771212709341025331457319097316152434053750514359997139905556
4096
2
3:
67100672
4913
2
2:
48275138
4913
2
9/4:
57407278.43822621313409295193909218492010086945005899615039500430477917121557264351931714127791271072
4913
2
3:
96540450
5832
2
2:
68024448
5832
2
9/4:
80892950.64394322732290691451773600736402320764276964152710018620536454710131697978018491949730193736
5832
2
3:
136037232
6859
2
2:
94091762
6859
2
9/4:
111891997.2903390526343758978003871578198725014055830829899678641332558728699853145506784016458380757
6859
2
3:
188169806
8000
2
2:
128000000
8000
2
9/4:
152215483.4065082530027725162322119495428557402818892077878239807284962220663537657889020795993690120
8000
2
3:
255984000
9261
2
2:
171532242
9261
2
9/4:
203983858.1445845002750866090507177092023025334346532899076655022499116846836043662212724277083376743
9261
2
3:
343045962
10648
2
2:
226759808
10648
2
9/4:
269660347.7155298506185786676849261150453665363027723740676179921575814897873688751763670743379366634
10648
2
3:
453498320
12167
2
2:
296071778
12167
2
9/4:
352086060.7831696248466693363545041421460448991372448076069679320585117241717748371794841873333990188
12167
2
3:
592119222
13824
2
2:
382205952
13824
2
9/4:
454516806.3164728926631649858688526147395157374171529202732697900155126470474938873079120734104251195
13824
2
3:
764384256
15625
2
2:
488281250
15625
2
9/4:
580661623.9000785608248193606261523656417936241454588519065278362192796007617923033069089914707457192
15625
2
3:
976531250
17576
2
2:
617831552
17576
2
9/4:
734723026.5030670652222312988123680027547031040246135280204527060236775175505838044663931798428211175
17576
2
3:
1235627952
19683
2
2:
774840978
19683
2
9/4:
921438955.7059776668790785258002092826561702295119787832807927103442290223224845972583430974476764651
19683
2
3:
1549642590
21952
2
2:
963780608
21952
2
9/4:
1146126449.386071351869687520748050091275832118755062632880739893419806131474870667734907878889873204
21952
2
3:
1927517312
24389
2
2:
1189646642
24389
2
9/4:
1414727021.860839335154939456398805031655964535406567691813410837937525547741937320687228003473616643
24389
2
3:
2379244506
27000
2
2:
1458000000
27000
2
9/4:
1733853756.489757168336512212078763618797727490319252742041402217520847551074760965564966178143999669
27000
2
3:
2915946000
29791
2
2:
1775007362
29791
2
9/4:
2110840110.734284451329458459896382081592728865120607447563421096580878485553363149156548434353382589
29791
2
3:
3549955142
32768
2
2:
2147483648
32768
2
9/4:
2553790433.676110147953119824141032771839906057667340217377989977530876842330776835030115438852909115
32768
2
3:
4294901760
35937
2
2:
2582935938
35937
2
9/4:
3071632195.993643505440377113881711180347725649379679216344226595363962938609114929735184018409158475
35937
2
3:
5165800002
39304
2
2:
3089608832
39304
2
9/4:
3674169932.396750577865236628765700560121701094455486523939698459594032192647641055765018898445933664
39304
2
3:
6179139056
42875
2
2:
3676531250
42875
2
9/4:
4372140896.519736353488752538017194156637228430964185440915352143559871994802842571279714655611185829
42875
2
3:
7352976750
46656
2
2:
4353564672
46656
2
9/4:
5177272428.272572486023285332635875045197740013820500943847517321092482174600505836589987884269074790
46656
2
3:
8707036032
50653
2
2:
5131452818
50653
2
9/4:
6102341033.650370629815096350795453575378176269638013287586985550545599063839793727401679576917165694
50653
2
3:
10262804330
54872
2
2:
6021872768
54872
2
9/4:
7161233177.001101378945278376442162422553775473462524755605163806189423155729325394820967718528761801
54872
2
3:
12043635792
59319
2
2:
7037487522
59319
2
9/4:
8369007785.751558810249022311093209246514181547385239558237302756967550360055258272120290094820373416
59319
2
3:
14074856406
64000
2
2:
8192000000
64000
2
9/4:
9741960467.591570630253219918835186957162869881035756878822799792617106854381372328264977314444322941
64000
2
3:
16383872000
Definition
For the wreath product $W_{q,n}=C_q\wr S_n=C_q^n\rtimes S_n$, where $C_q$ is cyclic of order $q$ and $S_n$ permutes the $n$ coordinates, let $D_{q,n}(s)=\sum_{\chi\in\operatorname{Irr}_{\mathbb C}(W_{q,n})}\chi(1)^s$ [7].
Parameters
$q$
—   order of the abelian base group ($q\geq1$)
$n$
—   permutation degree ($n\geq2$)
$s$
—   power-sum exponent ($s\in\mathbb{Q}$)
Formulas
(1)
Let $\Lambda=(\lambda^{(1)},\ldots,\lambda^{(q)})$ range over ordered $q$-tuples of partitions with $\sum_{a=1}^q|\lambda^{(a)}|=n$. For a partition $\lambda$, write $\lambda^{\prime}$ for its conjugate and $h_\lambda(i,j)=\lambda_i-j+\lambda_j^{\prime}-i+1$ for the hook length of its cell $(i,j)$, with rows and columns numbered from $1$. Then $d_{\Lambda}=n!\big/\prod_{a=1}^q\prod_{(i,j)\in\lambda^{(a)}}h_{\lambda^{(a)}}(i,j)$ and $D_{q,n}(s)=\sum_{\Lambda}d_{\Lambda}^{\,s}$; an empty partition contributes the product $1$ [1], Theorem 4.1.
(2)
$D_{q,n}(2)=|W_{q,n}|=q^n n!$, by the regular representation decomposition [2], Section 1.3.
(3)
$D_{q,2}(s)=2q+\binom{q}{2}2^s$, and in particular $D_{q,2}(3)=4q^2-2q$. There are $2q$ linear characters and $\binom{q}{2}$ characters of degree $2$.
(4)
$D_{q,n}(s)=\zeta_{W_{q,n}}(-s)$, with the finite representation zeta convention specified in the comments.
(5)
$D_{q,n}(0)=[z^n]\prod_{j\geq1}(1-z^j)^{-q}$, since irreducible characters are indexed by ordered $q$-tuples of partitions [1].
(6)
$D_{q,n}(1)=n!\sum_{j=0}^{\lfloor n/2\rfloor}\frac{q^{n-j}}{2^j j!(n-2j)!}$, obtained by counting the symmetric monomial matrices in the Gelfand model of [5].
(7)
For $p_\chi=\chi(1)^2/(q^n n!)$, $\sum_\chi p_\chi^2=D_{q,n}(4)/(q^n n!)^2$.
(8)
For a closed connected orientable surface $\Sigma_g$ of genus $g\geq1$, $|\operatorname{Hom}(\pi_1(\Sigma_g),W_{q,n})|=(q^n n!)^{2g-1}D_{q,n}(2-2g)$ [6], Section 3.
Comments
(9)
Each inequivalent irreducible complex character contributes once, including the trivial character and every other linear character. For every rational exponent, powers are taken on the positive real character degree. The sum is unnormalized: it is neither an average over characters nor an average with Plancherel weights. Equal degrees retain their full multiplicities.
(10)
For any finite abelian group $H$ of order $q$, the group $H\wr S_n$ has the same character-degree multiset as $W_{q,n}$. The colored-partition construction depends only on the $q$ degree-one characters of $H$ [1]. Equality of these multisets does not imply that the groups are isomorphic.
(11)
The cases $q=1$ and $q=2$ are the symmetric groups and the signed-permutation, or hyperoctahedral, groups [1]. Their Coxeter types are $A_{n-1}$ and $B_n$, respectively; Poincaré polynomials record a different invariant of these groups. The $q=3$ rows give an explicitly exploratory reference slice beyond these two specializations.
(12)
Over $\mathbb C$, a finite group algebra decomposes into matrix blocks whose sizes are the irreducible character degrees [2], Section 1.3. Consequently $D_{q,n}(3)$ is the sum-of-cubes cost appearing in the group-theoretic approach to matrix multiplication [3], Section 2. The comparison argument $s=9/4$ is chosen from the upper bound for the matrix-multiplication exponent in [4], Theorem 1.1. This choice does not assert that the exponent equals $9/4$ or that a character sum alone gives an algorithm.
(13)
In the two-factor construction of [3], Section 2, the abelian base is $H_m=(C_m)^3$. The wreath product $H_m\wr S_2$ therefore has character-degree sum $D_{m^3,2}(s)$. The cyclic factor order is denoted $m$ here and $n$ in that source; the parameter $n$ in this table always denotes the permutation degree.
(14)
For a finite group $G$, define the finite representation zeta function by $\zeta_G(z)=\sum_{\chi\in\operatorname{Irr}_{\mathbb C}(G)}\chi(1)^{-z}$. It is a finite sum, so every rational power-sum argument is allowed; $s=-2,-4$ gives zeta arguments $z=2,4$. Witten zeta values instead sum over infinitely many irreducible representations of a simple Lie algebra.
(15)
For positive integral $s$, the sum is a specialization of a power-sum symmetric polynomial with one variable per irreducible character, including repeated degrees. The hook factors in (1) also occur in the character theory associated with Schur polynomials; a symmetric-group character degree is not, in general, the value of a Schur polynomial at a fixed number of ones.
(16)
These character degrees describe the semisimple group algebra over $\mathbb C$. The augmentation Hilbert series of finite p-groups and low-degree growth constants concern group algebras in characteristic $p$ and are different invariants.
(17)
The zeroth moment counts irreducible characters, equivalently conjugacy classes. The first moment is the dimension of a Gelfand model: a representation containing each irreducible representation once. For the cyclic-base realization, it counts symmetric monomial matrices with nonzero entries among the $q$th roots of unity [5]. These matrices are ordinary involutions when $q=1,2$, but not generally when $q>2$. The second moment is the group order.
(18)
Under Plancherel measure, an irreducible character $\chi$ has probability $p_\chi=\chi(1)^2/|W_{q,n}|$. The fourth moment, divided by the squared group order, is the probability that two independent Plancherel samples are the same irreducible character. The table stores the unnormalized moment, not this probability.
(19)
Negative even arguments $s=2-2g$ occur in counting homomorphisms from the fundamental group of a closed connected orientable surface of genus $g$ to $W_{q,n}$ [6]. The arguments $-2$ and $-4$ correspond to genera $2$ and $3$. The entries are exact rational sums; multiplying by $|W_{q,n}|^{2g-1}$ gives the integer homomorphism count, without quotienting by conjugation.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from sage.combinat.partition import Partitions  # initialize partition support first
from sage.combinat.partition_tuple import PartitionTuples
from math import factorial

def character_degree_sum(q, n, s, digits=100):
    q, n, s = int(q), int(n), QQ(s)
    if q < 1 or n < 2:
        raise ValueError('q>=1 and n>=2 are required; s may be any rational')
    if n == 2:
        degrees = [(1, 2*q), (2, q*(q-1)//2)]
    else:
        degrees = []
        for shapes in PartitionTuples(q, n):
            hooks = ZZ(1)
            for shape in ([shapes] if q == 1 else shapes):
                hooks *= shape.hook_product(1)
            degrees.append((ZZ(factorial(n)) // hooks, 1))
    if s.denominator() == 1:
        return sum(QQ(count)*QQ(degree)**int(s) for degree, count in degrees)
    R = RealBallField(numberdb.bits(digits, losing=64))
    return sum(R(count)*R(d)**R(s) for d, count in degrees)

# One further signed-permutation group.
character_degree_sum(2, 13, '9/4')
References
[1]
H. Gao and N. Jing, Irreducible Characters of the Generalized Symmetric Group, Journal of Lie Theory 35 (2025), 593–616. Theorem 4.1, equation (4.4). (doi)
[2]
H. Cohn and C. Umans, A Group-theoretic Approach to Fast Matrix Multiplication (2003). Sections 1.3, 4 and 7.2. (arXiv)
[3]
H. Cohn, R. Kleinberg, B. Szegedy and C. Umans, Group-theoretic Algorithms for Matrix Multiplication (2005). Theorem 1.8 and Section 2. (arXiv)
[4]
A. Amanov, M. van den Berg and J. Zuiddam, Multiplying matrices via polynomials: exposition and barriers (2026). Theorem 1.1. (arXiv)
[5]
E. Marberg, Generalized Involution Models for Wreath Products, Israel Journal of Mathematics 192 (2012), 157–195. Introduction and Section 5. (arXiv) (doi)
[6]
M. R. Klug, Counting homomorphisms from surface groups to finite groups, Canadian Mathematical Bulletin 68 (2025), 141–153. Corollary 1 and Section 3. (arXiv) (doi)
Links
Similar tables
Matrix multiplication complexity —   records the complexity exponent that motivates the power-sum arguments, rather than a character-degree cost
Degrees of the irreducible characters of the sporadic simple groups —   records individual nontrivial character degrees for another group family; the sums here include all linear characters
Witten zeta values of simple Lie algebras —   records infinite representation-dimension zeta sums; the negative even power-sum arguments here are their finite-group analogues at positive even zeta arguments
Power sum symmetric polynomials —   gives formal power sums whose specialization to small full character-degree multisets recovers the corresponding integer moments
Schur polynomials —   provides the symmetric-function character context of the partition hook formula
Poincaré polynomials of the finite Coxeter groups —   its types A and B have the same underlying groups as the q=1 and q=2 slices, with group order obtained by evaluation at 1
Augmentation Hilbert series of finite p-groups —   records graded dimensions of modular group algebras, in contrast to the characteristic-zero irreducible block dimensions used here
Low-degree growth constants of finite p-group algebras —   describes direct-power growth for the modular augmentation filtration, rather than moments of complex character degrees
Ellenberg–Gijswijt constants and reduced polynomial multiplication subrank —   gives constants in sum-free bounds that constrain related group-theoretic approaches, rather than character-degree sums
Matrix multiplication bounds from the Cohn-Kleinberg-Szegedy-Umans wreath-product construction —   uses the permutation-degree-two cost with $q=m^3$: for $H_m=(C_m)^3$, its construction bound $\beta_m$ is the unique root in $(2,3)$ of $[2m(m-1)]^{\beta_m}=D_{m^3,2}(\beta_m)$ for $m\geq5$, whereas the moments here evaluate the same cost at the fixed arguments $s=2,9/4,3$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $s=0,1,2,9/4,3,4,-2,-4$ for $q=1$, $3\leq n\leq20$; $q=2$, $2\leq n\leq12$; and $q=3$, $2\leq n\leq10$. These are the symmetric and signed-permutation families and a noncanonical three-colour reference beyond them. The additional $q=m^3$, $n=2$ slice has $2\leq m\leq40$ at $s=2,9/4,3$, selected by the two-factor matrix-multiplication construction of [3]. The arguments cover character counts, Gelfand dimensions, group orders, matrix-block costs, the fourth Plancherel-related moment and the genus-2 and genus-3 surface sums, not an interpolation grid. The $9/4$ comparison comes from [4]. The finite group ranges are practical cutoffs. There are 77 group pairs and 421 values; all 154 earlier entries are retained)
How they were obtained:

Integral exponents are computed using exact rational arithmetic. Nonnegative integral exponents give integers; negative integral exponents are stored as reduced fractions, not rounded decimals. The $9/4$ sums use exact integer character degrees and Arb enclosures of $\sum_d a_d d^2\sqrt{\sqrt d}$, where $a_d$ is the degree multiplicity. These finite enclosures justify the 100 significant digits recorded. The computation uses 64 extra working bits and retains at least 391 bits of relative accuracy.

more

Independent coloured Young-graph path counts reproduce all 38 small-base character-degree spectra. All 421 values are checked against these spectra or the exact two-factor orbit formula. Integer fourth-root isolation gives rational bounds contained in every displayed $9/4$ interval. The group-order and Gelfand-dimension formulas pass for all 77 group pairs; coloured-partition counts check all 38 small-base cases. Direct enumeration and convolution of commutators in four small groups verifies the orientable-surface formula for genera 1, 2 and 3. The attached coverage-evidence.json records the checks for this revision. Earlier numerical and repair reports retain their historical 154-entry scope. All original numerical strings are preserved and have been reverified, not regenerated with different rounding.



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