Poincaré polynomials of the finite Coxeter groups

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Polynomials

type

$W(t)$

A1:

t + 1

A2:

t^3 + 2*t^2 + 2*t + 1

A3:

t^6 + 3*t^5 + 5*t^4 + 6*t^3 + 5*t^2 + 3*t + 1

A4:

t^10 + 4*t^9 + 9*t^8 + 15*t^7 + 20*t^6 + 22*t^5 + 20*t^4 + 15*t^3 + 9*t^2 + 4*t + 1

A5:

t^15 + 5*t^14 + 14*t^13 + 29*t^12 + 49*t^11 + 71*t^10 + 90*t^9 + 101*t^8 + 101*t^7 + 90*t^6 + 71*t^5 + 49*t^4 + 29*t^3 + 14*t^2 + 5*t + 1

A6:

t^21 + 6*t^20 + 20*t^19 + 49*t^18 + 98*t^17 + 169*t^16 + 259*t^15 + 359*t^14 + 455*t^13 + 531*t^12 + 573*t^11 + 573*t^10 + 531*t^9 + 455*t^8 + 359*t^7 + 259*t^6 + 169*t^5 + 98*t^4 + 49*t^3 + 20*t^2 + 6*t + 1

A7:

t^28 + 7*t^27 + 27*t^26 + 76*t^25 + 174*t^24 + 343*t^23 + 602*t^22 + 961*t^21 + 1415*t^20 + 1940*t^19 + 2493*t^18 + 3017*t^17 + 3450*t^16 + 3736*t^15 + 3836*t^14 + 3736*t^13 + 3450*t^12 + 3017*t^11 + 2493*t^10 + 1940*t^9 + 1415*t^8 + 961*t^7 + 602*t^6 + 343*t^5 + 174*t^4 + 76*t^3 + 27*t^2 + 7*t + 1

A8:

t^36 + 8*t^35 + 35*t^34 + 111*t^33 + 285*t^32 + 628*t^31 + 1230*t^30 + 2191*t^29 + 3606*t^28 + 5545*t^27 + 8031*t^26 + 11021*t^25 + 14395*t^24 + 17957*t^23 + 21450*t^22 + 24584*t^21 + 27073*t^20 + 28675*t^19 + 29228*t^18 + 28675*t^17 + 27073*t^16 + 24584*t^15 + 21450*t^14 + 17957*t^13 + 14395*t^12 + 11021*t^11 + 8031*t^10 + 5545*t^9 + 3606*t^8 + 2191*t^7 + 1230*t^6 + 628*t^5 + 285*t^4 + 111*t^3 + 35*t^2 + 8*t + 1

A9:

t^45 + 9*t^44 + 44*t^43 + 155*t^42 + 440*t^41 + 1068*t^40 + 2298*t^39 + 4489*t^38 + 8095*t^37 + 13640*t^36 + 21670*t^35 + 32683*t^34 + 47043*t^33 + 64889*t^32 + 86054*t^31 + 110010*t^30 + 135853*t^29 + 162337*t^28 + 187959*t^27 + 211089*t^26 + 230131*t^25 + 243694*t^24 + 250749*t^23 + 250749*t^22 + 243694*t^21 + 230131*t^20 + 211089*t^19 + 187959*t^18 + 162337*t^17 + 135853*t^16 + 110010*t^15 + 86054*t^14 + 64889*t^13 + 47043*t^12 + 32683*t^11 + 21670*t^10 + 13640*t^9 + 8095*t^8 + 4489*t^7 + 2298*t^6 + 1068*t^5 + 440*t^4 + 155*t^3 + 44*t^2 + 9*t + 1

B2:

t^4 + 2*t^3 + 2*t^2 + 2*t + 1

B3:

t^9 + 3*t^8 + 5*t^7 + 7*t^6 + 8*t^5 + 8*t^4 + 7*t^3 + 5*t^2 + 3*t + 1

B4:

t^16 + 4*t^15 + 9*t^14 + 16*t^13 + 24*t^12 + 32*t^11 + 39*t^10 + 44*t^9 + 46*t^8 + 44*t^7 + 39*t^6 + 32*t^5 + 24*t^4 + 16*t^3 + 9*t^2 + 4*t + 1

B5:

t^25 + 5*t^24 + 14*t^23 + 30*t^22 + 54*t^21 + 86*t^20 + 125*t^19 + 169*t^18 + 215*t^17 + 259*t^16 + 297*t^15 + 325*t^14 + 340*t^13 + 340*t^12 + 325*t^11 + 297*t^10 + 259*t^9 + 215*t^8 + 169*t^7 + 125*t^6 + 86*t^5 + 54*t^4 + 30*t^3 + 14*t^2 + 5*t + 1

B6:

t^36 + 6*t^35 + 20*t^34 + 50*t^33 + 104*t^32 + 190*t^31 + 315*t^30 + 484*t^29 + 699*t^28 + 958*t^27 + 1255*t^26 + 1580*t^25 + 1919*t^24 + 2254*t^23 + 2565*t^22 + 2832*t^21 + 3037*t^20 + 3166*t^19 + 3210*t^18 + 3166*t^17 + 3037*t^16 + 2832*t^15 + 2565*t^14 + 2254*t^13 + 1919*t^12 + 1580*t^11 + 1255*t^10 + 958*t^9 + 699*t^8 + 484*t^7 + 315*t^6 + 190*t^5 + 104*t^4 + 50*t^3 + 20*t^2 + 6*t + 1

B7:

t^49 + 7*t^48 + 27*t^47 + 77*t^46 + 181*t^45 + 371*t^44 + 686*t^43 + 1170*t^42 + 1869*t^41 + 2827*t^40 + 4082*t^39 + 5662*t^38 + 7581*t^37 + 9835*t^36 + 12399*t^35 + 15225*t^34 + 18242*t^33 + 21358*t^32 + 24464*t^31 + 27440*t^30 + 30162*t^29 + 32510*t^28 + 34376*t^27 + 35672*t^26 + 36336*t^25 + 36336*t^24 + 35672*t^23 + 34376*t^22 + 32510*t^21 + 30162*t^20 + 27440*t^19 + 24464*t^18 + 21358*t^17 + 18242*t^16 + 15225*t^15 + 12399*t^14 + 9835*t^13 + 7581*t^12 + 5662*t^11 + 4082*t^10 + 2827*t^9 + 1869*t^8 + 1170*t^7 + 686*t^6 + 371*t^5 + 181*t^4 + 77*t^3 + 27*t^2 + 7*t + 1

B8:

t^64 + 8*t^63 + 35*t^62 + 112*t^61 + 293*t^60 + 664*t^59 + 1350*t^58 + 2520*t^57 + 4389*t^56 + 7216*t^55 + 11298*t^54 + 16960*t^53 + 24541*t^52 + 34376*t^51 + 46775*t^50 + 62000*t^49 + 80241*t^48 + 101592*t^47 + 126029*t^46 + 153392*t^45 + 183373*t^44 + 215512*t^43 + 249202*t^42 + 283704*t^41 + 318171*t^40 + 351680*t^39 + 383270*t^38 + 411984*t^37 + 436913*t^36 + 457240*t^35 + 472281*t^34 + 481520*t^33 + 484636*t^32 + 481520*t^31 + 472281*t^30 + 457240*t^29 + 436913*t^28 + 411984*t^27 + 383270*t^26 + 351680*t^25 + 318171*t^24 + 283704*t^23 + 249202*t^22 + 215512*t^21 + 183373*t^20 + 153392*t^19 + 126029*t^18 + 101592*t^17 + 80241*t^16 + 62000*t^15 + 46775*t^14 + 34376*t^13 + 24541*t^12 + 16960*t^11 + 11298*t^10 + 7216*t^9 + 4389*t^8 + 2520*t^7 + 1350*t^6 + 664*t^5 + 293*t^4 + 112*t^3 + 35*t^2 + 8*t + 1

B9:

t^81 + 9*t^80 + 44*t^79 + 156*t^78 + 449*t^77 + 1113*t^76 + 2463*t^75 + 4983*t^74 + 9372*t^73 + 16588*t^72 + 27886*t^71 + 44846*t^70 + 69387*t^69 + 103763*t^68 + 150538*t^67 + 212538*t^66 + 292779*t^65 + 394371*t^64 + 520399*t^63 + 673783*t^62 + 857121*t^61 + 1072521*t^60 + 1321430*t^59 + 1604470*t^58 + 1921291*t^57 + 2270451*t^56 + 2649332*t^55 + 3054100*t^54 + 3479715*t^53 + 3919995*t^52 + 4367735*t^51 + 4814879*t^50 + 5252740*t^49 + 5672260*t^48 + 6064300*t^47 + 6419948*t^46 + 6730832*t^45 + 6989424*t^44 + 7189321*t^43 + 7325489*t^42 + 7394458*t^41 + 7394458*t^40 + 7325489*t^39 + 7189321*t^38 + 6989424*t^37 + 6730832*t^36 + 6419948*t^35 + 6064300*t^34 + 5672260*t^33 + 5252740*t^32 + 4814879*t^31 + 4367735*t^30 + 3919995*t^29 + 3479715*t^28 + 3054100*t^27 + 2649332*t^26 + 2270451*t^25 + 1921291*t^24 + 1604470*t^23 + 1321430*t^22 + 1072521*t^21 + 857121*t^20 + 673783*t^19 + 520399*t^18 + 394371*t^17 + 292779*t^16 + 212538*t^15 + 150538*t^14 + 103763*t^13 + 69387*t^12 + 44846*t^11 + 27886*t^10 + 16588*t^9 + 9372*t^8 + 4983*t^7 + 2463*t^6 + 1113*t^5 + 449*t^4 + 156*t^3 + 44*t^2 + 9*t + 1

D4:

t^12 + 4*t^11 + 9*t^10 + 16*t^9 + 23*t^8 + 28*t^7 + 30*t^6 + 28*t^5 + 23*t^4 + 16*t^3 + 9*t^2 + 4*t + 1

D5:

t^20 + 5*t^19 + 14*t^18 + 30*t^17 + 54*t^16 + 85*t^15 + 120*t^14 + 155*t^13 + 185*t^12 + 205*t^11 + 212*t^10 + 205*t^9 + 185*t^8 + 155*t^7 + 120*t^6 + 85*t^5 + 54*t^4 + 30*t^3 + 14*t^2 + 5*t + 1

D6:

t^30 + 6*t^29 + 20*t^28 + 50*t^27 + 104*t^26 + 190*t^25 + 314*t^24 + 478*t^23 + 679*t^22 + 908*t^21 + 1151*t^20 + 1390*t^19 + 1605*t^18 + 1776*t^17 + 1886*t^16 + 1924*t^15 + 1886*t^14 + 1776*t^13 + 1605*t^12 + 1390*t^11 + 1151*t^10 + 908*t^9 + 679*t^8 + 478*t^7 + 314*t^6 + 190*t^5 + 104*t^4 + 50*t^3 + 20*t^2 + 6*t + 1

D7:

t^42 + 7*t^41 + 27*t^40 + 77*t^39 + 181*t^38 + 371*t^37 + 686*t^36 + 1169*t^35 + 1862*t^34 + 2800*t^33 + 4005*t^32 + 5481*t^31 + 7210*t^30 + 9149*t^29 + 11230*t^28 + 13363*t^27 + 15442*t^26 + 17353*t^25 + 18983*t^24 + 20230*t^23 + 21013*t^22 + 21280*t^21 + 21013*t^20 + 20230*t^19 + 18983*t^18 + 17353*t^17 + 15442*t^16 + 13363*t^15 + 11230*t^14 + 9149*t^13 + 7210*t^12 + 5481*t^11 + 4005*t^10 + 2800*t^9 + 1862*t^8 + 1169*t^7 + 686*t^6 + 371*t^5 + 181*t^4 + 77*t^3 + 27*t^2 + 7*t + 1

D8:

t^56 + 8*t^55 + 35*t^54 + 112*t^53 + 293*t^52 + 664*t^51 + 1350*t^50 + 2520*t^49 + 4388*t^48 + 7208*t^47 + 11263*t^46 + 16848*t^45 + 24248*t^44 + 33712*t^43 + 45425*t^42 + 59480*t^41 + 75853*t^40 + 94384*t^39 + 114766*t^38 + 136544*t^37 + 159125*t^36 + 181800*t^35 + 203777*t^34 + 224224*t^33 + 242318*t^32 + 257296*t^31 + 268504*t^30 + 275440*t^29 + 277788*t^28 + 275440*t^27 + 268504*t^26 + 257296*t^25 + 242318*t^24 + 224224*t^23 + 203777*t^22 + 181800*t^21 + 159125*t^20 + 136544*t^19 + 114766*t^18 + 94384*t^17 + 75853*t^16 + 59480*t^15 + 45425*t^14 + 33712*t^13 + 24248*t^12 + 16848*t^11 + 11263*t^10 + 7208*t^9 + 4388*t^8 + 2520*t^7 + 1350*t^6 + 664*t^5 + 293*t^4 + 112*t^3 + 35*t^2 + 8*t + 1

D9:

t^72 + 9*t^71 + 44*t^70 + 156*t^69 + 449*t^68 + 1113*t^67 + 2463*t^66 + 4983*t^65 + 9372*t^64 + 16587*t^63 + 27877*t^62 + 44802*t^61 + 69231*t^60 + 103314*t^59 + 149425*t^58 + 210075*t^57 + 287796*t^56 + 384999*t^55 + 503812*t^54 + 645906*t^53 + 812319*t^52 + 1003290*t^51 + 1218116*t^50 + 1455045*t^49 + 1711216*t^48 + 1982655*t^47 + 2264333*t^46 + 2550288*t^45 + 2833809*t^44 + 3107676*t^43 + 3364445*t^42 + 3596763*t^41 + 3797695*t^40 + 3961044*t^39 + 4081645*t^38 + 4155615*t^37 + 4180544*t^36 + 4155615*t^35 + 4081645*t^34 + 3961044*t^33 + 3797695*t^32 + 3596763*t^31 + 3364445*t^30 + 3107676*t^29 + 2833809*t^28 + 2550288*t^27 + 2264333*t^26 + 1982655*t^25 + 1711216*t^24 + 1455045*t^23 + 1218116*t^22 + 1003290*t^21 + 812319*t^20 + 645906*t^19 + 503812*t^18 + 384999*t^17 + 287796*t^16 + 210075*t^15 + 149425*t^14 + 103314*t^13 + 69231*t^12 + 44802*t^11 + 27877*t^10 + 16587*t^9 + 9372*t^8 + 4983*t^7 + 2463*t^6 + 1113*t^5 + 449*t^4 + 156*t^3 + 44*t^2 + 9*t + 1

E6:

t^36 + 6*t^35 + 20*t^34 + 50*t^33 + 105*t^32 + 195*t^31 + 329*t^30 + 514*t^29 + 754*t^28 + 1048*t^27 + 1389*t^26 + 1765*t^25 + 2159*t^24 + 2549*t^23 + 2911*t^22 + 3222*t^21 + 3461*t^20 + 3611*t^19 + 3662*t^18 + 3611*t^17 + 3461*t^16 + 3222*t^15 + 2911*t^14 + 2549*t^13 + 2159*t^12 + 1765*t^11 + 1389*t^10 + 1048*t^9 + 754*t^8 + 514*t^7 + 329*t^6 + 195*t^5 + 105*t^4 + 50*t^3 + 20*t^2 + 6*t + 1

E7:

t^63 + 7*t^62 + 27*t^61 + 77*t^60 + 182*t^59 + 378*t^58 + 713*t^57 + 1247*t^56 + 2051*t^55 + 3205*t^54 + 4795*t^53 + 6909*t^52 + 9632*t^51 + 13040*t^50 + 17194*t^49 + 22134*t^48 + 27874*t^47 + 34398*t^46 + 41657*t^45 + 49567*t^44 + 58009*t^43 + 66831*t^42 + 75852*t^41 + 84868*t^40 + 93659*t^39 + 101997*t^38 + 109655*t^37 + 116417*t^36 + 122087*t^35 + 126497*t^34 + 129514*t^33 + 131046*t^32 + 131046*t^31 + 129514*t^30 + 126497*t^29 + 122087*t^28 + 116417*t^27 + 109655*t^26 + 101997*t^25 + 93659*t^24 + 84868*t^23 + 75852*t^22 + 66831*t^21 + 58009*t^20 + 49567*t^19 + 41657*t^18 + 34398*t^17 + 27874*t^16 + 22134*t^15 + 17194*t^14 + 13040*t^13 + 9632*t^12 + 6909*t^11 + 4795*t^10 + 3205*t^9 + 2051*t^8 + 1247*t^7 + 713*t^6 + 378*t^5 + 182*t^4 + 77*t^3 + 27*t^2 + 7*t + 1

E8:

t^120 + 8*t^119 + 35*t^118 + 112*t^117 + 294*t^116 + 672*t^115 + 1386*t^114 + 2640*t^113 + 4718*t^112 + 8000*t^111 + 12978*t^110 + 20272*t^109 + 30645*t^108 + 45016*t^107 + 64470*t^106 + 90264*t^105 + 123829*t^104 + 166768*t^103 + 220849*t^102 + 287992*t^101 + 370250*t^100 + 469784*t^99 + 588833*t^98 + 729680*t^97 + 894613*t^96 + 1085880*t^95 + 1305640*t^94 + 1555912*t^93 + 1838523*t^92 + 2155056*t^91 + 2506798*t^90 + 2894688*t^89 + 3319268*t^88 + 3780640*t^87 + 4278429*t^86 + 4811752*t^85 + 5379194*t^84 + 5978792*t^83 + 6608029*t^82 + 7263840*t^81 + 7942628*t^80 + 8640288*t^79 + 9352240*t^78 + 10073472*t^77 + 10798593*t^76 + 11521896*t^75 + 12237428*t^74 + 12939064*t^73 + 13620586*t^72 + 14275768*t^71 + 14898464*t^70 + 15482696*t^69 + 16022740*t^68 + 16513208*t^67 + 16949127*t^66 + 17326016*t^65 + 17639957*t^64 + 17887656*t^63 + 18066494*t^62 + 18174568*t^61 + 18210722*t^60 + 18174568*t^59 + 18066494*t^58 + 17887656*t^57 + 17639957*t^56 + 17326016*t^55 + 16949127*t^54 + 16513208*t^53 + 16022740*t^52 + 15482696*t^51 + 14898464*t^50 + 14275768*t^49 + 13620586*t^48 + 12939064*t^47 + 12237428*t^46 + 11521896*t^45 + 10798593*t^44 + 10073472*t^43 + 9352240*t^42 + 8640288*t^41 + 7942628*t^40 + 7263840*t^39 + 6608029*t^38 + 5978792*t^37 + 5379194*t^36 + 4811752*t^35 + 4278429*t^34 + 3780640*t^33 + 3319268*t^32 + 2894688*t^31 + 2506798*t^30 + 2155056*t^29 + 1838523*t^28 + 1555912*t^27 + 1305640*t^26 + 1085880*t^25 + 894613*t^24 + 729680*t^23 + 588833*t^22 + 469784*t^21 + 370250*t^20 + 287992*t^19 + 220849*t^18 + 166768*t^17 + 123829*t^16 + 90264*t^15 + 64470*t^14 + 45016*t^13 + 30645*t^12 + 20272*t^11 + 12978*t^10 + 8000*t^9 + 4718*t^8 + 2640*t^7 + 1386*t^6 + 672*t^5 + 294*t^4 + 112*t^3 + 35*t^2 + 8*t + 1

F4:

t^24 + 4*t^23 + 9*t^22 + 16*t^21 + 25*t^20 + 36*t^19 + 48*t^18 + 60*t^17 + 71*t^16 + 80*t^15 + 87*t^14 + 92*t^13 + 94*t^12 + 92*t^11 + 87*t^10 + 80*t^9 + 71*t^8 + 60*t^7 + 48*t^6 + 36*t^5 + 25*t^4 + 16*t^3 + 9*t^2 + 4*t + 1

G2:

t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1

H3:

t^15 + 3*t^14 + 5*t^13 + 7*t^12 + 9*t^11 + 11*t^10 + 12*t^9 + 12*t^8 + 12*t^7 + 12*t^6 + 11*t^5 + 9*t^4 + 7*t^3 + 5*t^2 + 3*t + 1

H4:

t^60 + 4*t^59 + 9*t^58 + 16*t^57 + 25*t^56 + 36*t^55 + 49*t^54 + 64*t^53 + 81*t^52 + 100*t^51 + 121*t^50 + 144*t^49 + 168*t^48 + 192*t^47 + 216*t^46 + 240*t^45 + 264*t^44 + 288*t^43 + 312*t^42 + 336*t^41 + 359*t^40 + 380*t^39 + 399*t^38 + 416*t^37 + 431*t^36 + 444*t^35 + 455*t^34 + 464*t^33 + 471*t^32 + 476*t^31 + 478*t^30 + 476*t^29 + 471*t^28 + 464*t^27 + 455*t^26 + 444*t^25 + 431*t^24 + 416*t^23 + 399*t^22 + 380*t^21 + 359*t^20 + 336*t^19 + 312*t^18 + 288*t^17 + 264*t^16 + 240*t^15 + 216*t^14 + 192*t^13 + 168*t^12 + 144*t^11 + 121*t^10 + 100*t^9 + 81*t^8 + 64*t^7 + 49*t^6 + 36*t^5 + 25*t^4 + 16*t^3 + 9*t^2 + 4*t + 1

I2(5):

t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1

I2(7):

t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1

I2(8):

t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1

I2(9):

t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1

I2(10):

t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1

I2(11):

t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1

I2(12):

t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1

Definition

For a finite irreducible Coxeter group $W$ [2], with simple-reflection length $\ell$, this table stores the length generating polynomial $W(t)=\sum_{w\in W}t^{\ell(w)}$.

Parameters

type

— Coxeter type (an irreducible finite Coxeter type $A_n$, $B_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, $G_2$, $H_3$, $H_4$ or $I_2(m)$)

Formulas

(1)

If $d_1,\ldots,d_r$ are the degrees of a homogeneous algebraically independent generating set for the $W$-invariant polynomial algebra, then $W(t)=\prod_{i=1}^r(1+t+\cdots+t^{d_i-1})$ [1].

(2)

$W(1)=\prod_i d_i=|W|$. In type $A_n$, this gives $W(1)=(n+1)!$.

(3)

If $N=\sum_i(d_i-1)$, then $t^N W(t^{-1})=W(t)$.

(4)

For simple reflections $S$, $\sum_{J\subseteq S}(-1)^{|J|}W(t)/W_J(t)=t^N$, where $W_J$ is the parabolic subgroup generated by $J$.

Comments

(5)

The variable is $t$, following [1]. In point-counting formulas for split reductive groups over $\mathbb F_q$, the same polynomial is evaluated at $q$.

(6)

In type $A_n$, the Coxeter group is the symmetric group on $n+1$ letters and $\ell$ is the inversion number, so the coefficients of $W(t)$ are the Mahonian numbers [4]. For each finite Coxeter group here, $W(t)$ is the Hilbert series of the coinvariant algebra, the quotient of the polynomial ring on the reflection representation by the ideal generated by positive-degree $W$-invariants [1].

(7)

For a split reductive group $G$ with Weyl group $W$ and Borel subgroup $B$, $W(q)$ is the number of $\mathbb F_q$-points of the flag variety $G/B$ [3]. The polynomial $W(t^2)$ is the ordinary Poincaré polynomial of the corresponding complex flag manifold.

(8)

$B_n$ and $C_n$ have the same Coxeter group, so there is no separate $C_n$ row. $I_2(3)=A_2$, $I_2(4)=B_2$ and $I_2(6)=G_2$, so those three dihedral types have no rows of their own either. Reducible finite Coxeter groups are not rows; their polynomials are products of the irreducible rows.

Programs

(P1)

Sage

from sage.all import ZZ, PolynomialRing, CoxeterGroup

T = PolynomialRing(ZZ, 't')
t = T.gen()

def poincare_from_degrees(degrees):
    out = T.one()
    for d in degrees:
        out *= sum(t**k for k in range(d))
    return out

print(poincare_from_degrees(CoxeterGroup(['F', 4]).degrees()))  # F4
print(poincare_from_degrees(CoxeterGroup(['I', 5]).degrees()))  # I2(5)

References

[1]

James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, 1990.

Links

[2]

Wikipedia: Coxeter group

[3]

Wikipedia: Flag variety

[4]

OEIS A008302: Mahonian triangle

Similar tables

Cyclotomic polynomials — each entry factors as a product of cyclotomic polynomials

Factorial of natural numbers — gives the values $W(1)=(n+1)!$ for type $A_n$

Orders of finite simple groups of Lie type — the degrees in the Weyl group polynomial also appear in the order formulas for the split groups of Lie type

Growth rates of hyperbolic Coxeter simplex groups — growth rates for infinite Coxeter groups rather than finite growth polynomials

Data properties

Entries are of type: integral polynomial

Table is complete: no (it holds $A_n$ for $1\leq n\leq9$, $B_n$ for $2\leq n\leq9$, $D_n$ for $4\leq n\leq9$, the exceptional types $E_6$, $E_7$, $E_8$, $F_4$, $G_2$, $H_3$ and $H_4$, and $I_2(m)$ for $5\leq m\leq12$ with $m\neq6$, a range chosen for small ranks and small dihedral parameters rather than for a size limit)

How well the digits are known: exact

How they were obtained:

The entries are exact integer polynomials computed from (1).

Checks for this table compare $W(1)$ with the known orders of $E_6$, $E_7$, $E_8$, $F_4$, $G_2$, $H_3$ and $H_4$, verify Solomon's identity (4) for all stored rows, and compare the type $A$ rows with the Mahonian triangle [4].