Augmentation Hilbert series of finite p-groups

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Polynomials

$G$

$H_G(t)$

$U_{3}(\mathbb F_{2})$:

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

comment: This group is the dihedral group of order $8$, also $\operatorname{Heis}_1(\mathbb F_2)$. Its Hilbert series equals the Poincaré polynomial of the Coxeter group $B_2$.

$U_{4}(\mathbb F_{2})$:

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

$U_{5}(\mathbb F_{2})$:

t^20 + 4*t^19 + 9*t^18 + 18*t^17 + 31*t^16 + 46*t^15 + 64*t^14 + 82*t^13 + 96*t^12 + 106*t^11 + 110*t^10 + 106*t^9 + 96*t^8 + 82*t^7 + 64*t^6 + 46*t^5 + 31*t^4 + 18*t^3 + 9*t^2 + 4*t + 1

$U_{6}(\mathbb F_{2})$:

t^35 + 5*t^34 + 14*t^33 + 33*t^32 + 68*t^31 + 124*t^30 + 210*t^29 + 332*t^28 + 492*t^27 + 693*t^26 + 931*t^25 + 1196*t^24 + 1476*t^23 + 1754*t^22 + 2008*t^21 + 2220*t^20 + 2374*t^19 + 2453*t^18 + 2453*t^17 + 2374*t^16 + 2220*t^15 + 2008*t^14 + 1754*t^13 + 1476*t^12 + 1196*t^11 + 931*t^10 + 693*t^9 + 492*t^8 + 332*t^7 + 210*t^6 + 124*t^5 + 68*t^4 + 33*t^3 + 14*t^2 + 5*t + 1

$\operatorname{Heis}_{2}(\mathbb F_{2})$:

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

$\operatorname{Heis}_{3}(\mathbb F_{2})$:

t^8 + 6*t^7 + 16*t^6 + 26*t^5 + 30*t^4 + 26*t^3 + 16*t^2 + 6*t + 1

$\operatorname{Heis}_{4}(\mathbb F_{2})$:

t^10 + 8*t^9 + 29*t^8 + 64*t^7 + 98*t^6 + 112*t^5 + 98*t^4 + 64*t^3 + 29*t^2 + 8*t + 1

$U_{3}(\mathbb F_{3})$:

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

$U_{4}(\mathbb F_{3})$:

t^20 + 3*t^19 + 8*t^18 + 14*t^17 + 24*t^16 + 34*t^15 + 47*t^14 + 57*t^13 + 67*t^12 + 72*t^11 + 75*t^10 + 72*t^9 + 67*t^8 + 57*t^7 + 47*t^6 + 34*t^5 + 24*t^4 + 14*t^3 + 8*t^2 + 3*t + 1

$U_{5}(\mathbb F_{3})$:

t^40 + 4*t^39 + 13*t^38 + 30*t^37 + 64*t^36 + 118*t^35 + 206*t^34 + 328*t^33 + 500*t^32 + 716*t^31 + 990*t^30 + 1304*t^29 + 1667*t^28 + 2048*t^27 + 2451*t^26 + 2832*t^25 + 3195*t^24 + 3492*t^23 + 3732*t^22 + 3870*t^21 + 3927*t^20 + 3870*t^19 + 3732*t^18 + 3492*t^17 + 3195*t^16 + 2832*t^15 + 2451*t^14 + 2048*t^13 + 1667*t^12 + 1304*t^11 + 990*t^10 + 716*t^9 + 500*t^8 + 328*t^7 + 206*t^6 + 118*t^5 + 64*t^4 + 30*t^3 + 13*t^2 + 4*t + 1

$U_{6}(\mathbb F_{3})$:

t^70 + 5*t^69 + 19*t^68 + 53*t^67 + 132*t^66 + 289*t^65 + 590*t^64 + 1114*t^63 + 1997*t^62 + 3393*t^61 + 5539*t^60 + 8684*t^59 + 13179*t^58 + 19365*t^57 + 27692*t^56 + 38548*t^55 + 52419*t^54 + 69653*t^53 + 90669*t^52 + 115650*t^51 + 144816*t^50 + 178057*t^49 + 215276*t^48 + 255966*t^47 + 299643*t^46 + 345378*t^45 + 392320*t^44 + 439184*t^43 + 484866*t^42 + 527898*t^41 + 567127*t^40 + 601134*t^39 + 628959*t^38 + 649482*t^37 + 662165*t^36 + 666383*t^35 + 662165*t^34 + 649482*t^33 + 628959*t^32 + 601134*t^31 + 567127*t^30 + 527898*t^29 + 484866*t^28 + 439184*t^27 + 392320*t^26 + 345378*t^25 + 299643*t^24 + 255966*t^23 + 215276*t^22 + 178057*t^21 + 144816*t^20 + 115650*t^19 + 90669*t^18 + 69653*t^17 + 52419*t^16 + 38548*t^15 + 27692*t^14 + 19365*t^13 + 13179*t^12 + 8684*t^11 + 5539*t^10 + 3393*t^9 + 1997*t^8 + 1114*t^7 + 590*t^6 + 289*t^5 + 132*t^4 + 53*t^3 + 19*t^2 + 5*t + 1

$\operatorname{Heis}_{2}(\mathbb F_{3})$:

t^12 + 4*t^11 + 11*t^10 + 20*t^9 + 30*t^8 + 36*t^7 + 39*t^6 + 36*t^5 + 30*t^4 + 20*t^3 + 11*t^2 + 4*t + 1

$\operatorname{Heis}_{3}(\mathbb F_{3})$:

t^16 + 6*t^15 + 22*t^14 + 56*t^13 + 112*t^12 + 182*t^11 + 252*t^10 + 302*t^9 + 321*t^8 + 302*t^7 + 252*t^6 + 182*t^5 + 112*t^4 + 56*t^3 + 22*t^2 + 6*t + 1

$\operatorname{Heis}_{4}(\mathbb F_{3})$:

t^20 + 8*t^19 + 37*t^18 + 120*t^17 + 303*t^16 + 624*t^15 + 1086*t^14 + 1632*t^13 + 2157*t^12 + 2536*t^11 + 2675*t^10 + 2536*t^9 + 2157*t^8 + 1632*t^7 + 1086*t^6 + 624*t^5 + 303*t^4 + 120*t^3 + 37*t^2 + 8*t + 1

$U_{3}(\mathbb F_{5})$:

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

$U_{4}(\mathbb F_{5})$:

t^40 + 3*t^39 + 8*t^38 + 17*t^37 + 33*t^36 + 55*t^35 + 88*t^34 + 129*t^33 + 182*t^32 + 243*t^31 + 316*t^30 + 393*t^29 + 479*t^28 + 563*t^27 + 648*t^26 + 725*t^25 + 796*t^24 + 851*t^23 + 896*t^22 + 921*t^21 + 931*t^20 + 921*t^19 + 896*t^18 + 851*t^17 + 796*t^16 + 725*t^15 + 648*t^14 + 563*t^13 + 479*t^12 + 393*t^11 + 316*t^10 + 243*t^9 + 182*t^8 + 129*t^7 + 88*t^6 + 55*t^5 + 33*t^4 + 17*t^3 + 8*t^2 + 3*t + 1

$U_{5}(\mathbb F_{5})$:

t^80 + 4*t^79 + 13*t^78 + 34*t^77 + 80*t^76 + 166*t^75 + 323*t^74 + 584*t^73 + 1005*t^72 + 1644*t^71 + 2593*t^70 + 3936*t^69 + 5804*t^68 + 8306*t^67 + 11603*t^66 + 15818*t^65 + 21130*t^64 + 27648*t^63 + 35549*t^62 + 44906*t^61 + 55856*t^60 + 68408*t^59 + 82639*t^58 + 98456*t^57 + 115855*t^56 + 134630*t^55 + 154673*t^54 + 175664*t^53 + 197404*t^52 + 219458*t^51 + 241563*t^50 + 263212*t^49 + 284100*t^48 + 303696*t^47 + 321713*t^46 + 337636*t^45 + 351248*t^44 + 362112*t^43 + 370111*t^42 + 374932*t^41 + 376599*t^40 + 374932*t^39 + 370111*t^38 + 362112*t^37 + 351248*t^36 + 337636*t^35 + 321713*t^34 + 303696*t^33 + 284100*t^32 + 263212*t^31 + 241563*t^30 + 219458*t^29 + 197404*t^28 + 175664*t^27 + 154673*t^26 + 134630*t^25 + 115855*t^24 + 98456*t^23 + 82639*t^22 + 68408*t^21 + 55856*t^20 + 44906*t^19 + 35549*t^18 + 27648*t^17 + 21130*t^16 + 15818*t^15 + 11603*t^14 + 8306*t^13 + 5804*t^12 + 3936*t^11 + 2593*t^10 + 1644*t^9 + 1005*t^8 + 584*t^7 + 323*t^6 + 166*t^5 + 80*t^4 + 34*t^3 + 13*t^2 + 4*t + 1

$\operatorname{Heis}_{2}(\mathbb F_{5})$:

t^24 + 4*t^23 + 11*t^22 + 24*t^21 + 46*t^20 + 76*t^19 + 114*t^18 + 156*t^17 + 199*t^16 + 236*t^15 + 266*t^14 + 284*t^13 + 291*t^12 + 284*t^11 + 266*t^10 + 236*t^9 + 199*t^8 + 156*t^7 + 114*t^6 + 76*t^5 + 46*t^4 + 24*t^3 + 11*t^2 + 4*t + 1

$\operatorname{Heis}_{3}(\mathbb F_{5})$:

t^32 + 6*t^31 + 22*t^30 + 62*t^29 + 148*t^28 + 308*t^27 + 574*t^26 + 974*t^25 + 1525*t^24 + 2220*t^23 + 3030*t^22 + 3900*t^21 + 4760*t^20 + 5530*t^19 + 6140*t^18 + 6530*t^17 + 6665*t^16 + 6530*t^15 + 6140*t^14 + 5530*t^13 + 4760*t^12 + 3900*t^11 + 3030*t^10 + 2220*t^9 + 1525*t^8 + 974*t^7 + 574*t^6 + 308*t^5 + 148*t^4 + 62*t^3 + 22*t^2 + 6*t + 1

$\operatorname{Heis}_{4}(\mathbb F_{5})$:

t^40 + 8*t^39 + 37*t^38 + 128*t^37 + 367*t^36 + 912*t^35 + 2019*t^34 + 4056*t^33 + 7494*t^32 + 12856*t^31 + 20633*t^30 + 31168*t^29 + 44537*t^28 + 60448*t^27 + 78207*t^26 + 96744*t^25 + 114720*t^24 + 130680*t^23 + 143245*t^22 + 151280*t^21 + 154045*t^20 + 151280*t^19 + 143245*t^18 + 130680*t^17 + 114720*t^16 + 96744*t^15 + 78207*t^14 + 60448*t^13 + 44537*t^12 + 31168*t^11 + 20633*t^10 + 12856*t^9 + 7494*t^8 + 4056*t^7 + 2019*t^6 + 912*t^5 + 367*t^4 + 128*t^3 + 37*t^2 + 8*t + 1

$U_{3}(\mathbb F_{7})$:

t^24 + 2*t^23 + 4*t^22 + 6*t^21 + 9*t^20 + 12*t^19 + 16*t^18 + 18*t^17 + 21*t^16 + 22*t^15 + 24*t^14 + 24*t^13 + 25*t^12 + 24*t^11 + 24*t^10 + 22*t^9 + 21*t^8 + 18*t^7 + 16*t^6 + 12*t^5 + 9*t^4 + 6*t^3 + 4*t^2 + 2*t + 1

$U_{4}(\mathbb F_{7})$:

t^60 + 3*t^59 + 8*t^58 + 17*t^57 + 33*t^56 + 58*t^55 + 97*t^54 + 150*t^53 + 224*t^52 + 318*t^51 + 438*t^50 + 582*t^49 + 756*t^48 + 954*t^47 + 1183*t^46 + 1434*t^45 + 1712*t^44 + 2006*t^43 + 2320*t^42 + 2638*t^41 + 2964*t^40 + 3282*t^39 + 3593*t^38 + 3882*t^37 + 4152*t^36 + 4386*t^35 + 4590*t^34 + 4750*t^33 + 4870*t^32 + 4940*t^31 + 4967*t^30 + 4940*t^29 + 4870*t^28 + 4750*t^27 + 4590*t^26 + 4386*t^25 + 4152*t^24 + 3882*t^23 + 3593*t^22 + 3282*t^21 + 2964*t^20 + 2638*t^19 + 2320*t^18 + 2006*t^17 + 1712*t^16 + 1434*t^15 + 1183*t^14 + 954*t^13 + 756*t^12 + 582*t^11 + 438*t^10 + 318*t^9 + 224*t^8 + 150*t^7 + 97*t^6 + 58*t^5 + 33*t^4 + 17*t^3 + 8*t^2 + 3*t + 1

$\operatorname{Heis}_{2}(\mathbb F_{7})$:

t^36 + 4*t^35 + 11*t^34 + 24*t^33 + 46*t^32 + 80*t^31 + 130*t^30 + 196*t^29 + 279*t^28 + 376*t^27 + 485*t^26 + 600*t^25 + 716*t^24 + 824*t^23 + 921*t^22 + 1000*t^21 + 1060*t^20 + 1096*t^19 + 1109*t^18 + 1096*t^17 + 1060*t^16 + 1000*t^15 + 921*t^14 + 824*t^13 + 716*t^12 + 600*t^11 + 485*t^10 + 376*t^9 + 279*t^8 + 196*t^7 + 130*t^6 + 80*t^5 + 46*t^4 + 24*t^3 + 11*t^2 + 4*t + 1

$\operatorname{Heis}_{3}(\mathbb F_{7})$:

t^48 + 6*t^47 + 22*t^46 + 62*t^45 + 148*t^44 + 314*t^43 + 610*t^42 + 1100*t^41 + 1861*t^40 + 2976*t^39 + 4528*t^38 + 6588*t^37 + 9204*t^36 + 12384*t^35 + 16094*t^34 + 20250*t^33 + 24725*t^32 + 29350*t^31 + 33930*t^30 + 38254*t^29 + 42120*t^28 + 45340*t^27 + 47760*t^26 + 49260*t^25 + 49769*t^24 + 49260*t^23 + 47760*t^22 + 45340*t^21 + 42120*t^20 + 38254*t^19 + 33930*t^18 + 29350*t^17 + 24725*t^16 + 20250*t^15 + 16094*t^14 + 12384*t^13 + 9204*t^12 + 6588*t^11 + 4528*t^10 + 2976*t^9 + 1861*t^8 + 1100*t^7 + 610*t^6 + 314*t^5 + 148*t^4 + 62*t^3 + 22*t^2 + 6*t + 1

$\operatorname{Heis}_{4}(\mathbb F_{7})$:

t^60 + 8*t^59 + 37*t^58 + 128*t^57 + 367*t^56 + 920*t^55 + 2083*t^54 + 4344*t^53 + 8454*t^52 + 15496*t^51 + 26942*t^50 + 44680*t^49 + 70994*t^48 + 108472*t^47 + 159845*t^46 + 227752*t^45 + 314454*t^44 + 421512*t^43 + 549472*t^42 + 697592*t^41 + 863674*t^40 + 1044008*t^39 + 1233463*t^38 + 1425704*t^37 + 1613542*t^36 + 1789368*t^35 + 1945650*t^34 + 2075416*t^33 + 2172716*t^32 + 2233000*t^31 + 2253419*t^30 + 2233000*t^29 + 2172716*t^28 + 2075416*t^27 + 1945650*t^26 + 1789368*t^25 + 1613542*t^24 + 1425704*t^23 + 1233463*t^22 + 1044008*t^21 + 863674*t^20 + 697592*t^19 + 549472*t^18 + 421512*t^17 + 314454*t^16 + 227752*t^15 + 159845*t^14 + 108472*t^13 + 70994*t^12 + 44680*t^11 + 26942*t^10 + 15496*t^9 + 8454*t^8 + 4344*t^7 + 2083*t^6 + 920*t^5 + 367*t^4 + 128*t^3 + 37*t^2 + 8*t + 1

Definition

For a nontrivial finite $p$-group $G$, let $I$ be the kernel of the augmentation map $\mathbb F_p[G]\to\mathbb F_p$ sending each group element to $1$. Its augmentation Hilbert series is $H_G(t)=\sum_{j\geq0}\dim_{\mathbb F_p}(I^j/I^{j+1})t^j$, with $I^0=\mathbb F_p[G]$ [4].

Parameters

$G$

— finite p-group (a nontrivial finite group of prime-power order $|G|=p^m$, with coefficients taken over the prime field $\mathbb F_p$)

Formulas

(1)

Jennings' product formula is $H_G(t)=\prod_{j\geq1}(1+t^j+\cdots+t^{(p-1)j})^{e_j}$ [1].

(2)

For $G=U_n(\mathbb F_p)$, $e_j=n-j$ for $1\leq j<n$ and $e_j=0$ otherwise. Thus $H_G(t)=\prod_{j=1}^{n-1}(1+t^j+\cdots+t^{(p-1)j})^{n-j}$ and $d_G:=\deg H_G=(p-1)n(n^2-1)/6$ [2].

(3)

For $G=\operatorname{Heis}_r(\mathbb F_p)$, $e_1=2r$, $e_2=1$, and the other $e_j$ vanish. Thus $H_G(t)=(1+t+\cdots+t^{p-1})^{2r}(1+t^2+\cdots+t^{2(p-1)})$ and $d_G=(p-1)(2r+2)$ [3].

(4)

$H_G(0)=1$ and $H_G(1)=|G|$. In particular, $|U_n(\mathbb F_p)|=p^{n(n-1)/2}$ and $|\operatorname{Heis}_r(\mathbb F_p)|=p^{2r+1}$.

(5)

Writing $d_G=\deg H_G$, the coefficients satisfy $t^{d_G}H_G(t^{-1})=H_G(t)$. Consequently $2H_G^{\prime}(1)=d_G|G|$.

Comments

(6)

The variable $t$ records augmentation degree. The coefficients are dimensions over $\mathbb F_p$, stored as integers in $\mathbb Z[t]$. Since the augmentation ideal of a finite $p$-group is nilpotent, the entire Hilbert series is a finite polynomial [2].

(7)

The group $U_n(\mathbb F_p)$ consists of upper unitriangular $n\times n$ matrices over the prime field, with $n\geq2$ and $p$ prime. Its row identity is $\mathrm{UT}\text{-}n\text{-}p$ [2].

(8)

The group $\operatorname{Heis}_r(\mathbb F_p)$ consists of triples $(a,b,c)$ with $a,b\in\mathbb F_p^r$ and $c\in\mathbb F_p$, where $(a,b,c)(a^{\prime},b^{\prime},c^{\prime})=(a+a^{\prime},b+b^{\prime},c+c^{\prime}+a\cdot b^{\prime})$. Here $r\geq1$, the row identity is $\mathrm{Heis}\text{-}r\text{-}p$, and $r=n-1$ in the notation of [3]. This multiplication law also specifies the group in characteristic two.

(9)

The rank-one Heisenberg group $\operatorname{Heis}_1(\mathbb F_p)$ is $U_3(\mathbb F_p)$, so it is represented once by the identity $\mathrm{UT}\text{-}3\text{-}p$. Rows are ordered by increasing $p$, then by unitriangular matrix size $n$, then by Heisenberg rank $r$.

(10)

For $j\geq1$, the dimension subgroup is $D_j(G)=\{g\in G:g-1\in I^j\}$, and $e_j=\log_p|D_j(G)/D_{j+1}(G)|$. This is the Jennings dimension-subgroup filtration, with $D_j(G)^p\subseteq D_{pj}(G)$ [1].

(11)

The value $H_{U_n(\mathbb F_p)}(1)$ is the $p$-part of the order of $\mathrm{SL}_n(\mathbb F_p)$, whose type $A_{n-1}$ order polynomial is in Orders of finite groups of Lie type.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
R = PolynomialRing(ZZ, 't')
t = R.gen()
def augmentation_hilbert(p, degrees):
    H = R.one()
    for j, e in degrees:
        H *= sum((t**(j*k) for k in range(p)), R.zero())**e
    return H
# U_4(F_3): e_j = 4-j for 1 <= j < 4.
print(augmentation_hilbert(3, [(j, 4-j) for j in range(1, 4)]))
# Heis_2(F_3): four generators of weight 1, one of weight 2.
print(augmentation_hilbert(3, [(1, 4), (2, 1)]))

References

[1]

Laurent Bartholdi and Rostislav I. Grigorchuk, Lie Methods in Growth of Groups and Groups of Finite Width, 2000, Section 2, equation (3). (arXiv)

[2]

Jonah Blasiak, Thomas Church, Henry Cohn, Joshua A. Grochow and Chris Umans, Which groups are amenable to proving exponent two for matrix multiplication?, 2017, Section 3.1.1 and Example 3.15. (arXiv)

[3]

Fedor Petrov, Combinatorial results implied by many zero divisors in a group ring, 2020, Section 7. (arXiv)

Links

[4]

Jennings theorem and dimension subgroups

Similar tables

Orders of finite groups of Lie type — the $p$-part of the type $A_{n-1}$ order at $q=p$ equals $H_{U_n(\mathbb F_p)}(1)$

Poincaré polynomials of the finite Coxeter groups — a related graded counting polynomial, with Coxeter length in place of augmentation degree; the $B_2$ polynomial equals $H_{U_3(\mathbb F_2)}(t)$

Low-degree growth constants of finite p-group algebras — for the same group $G$ over $\mathbb F_p$, stores $C_G=\min_{0<t\leq1}H_G(t)t^{-d_G/3}$, where $d_G=\deg H_G$; equivalently, $C_G$ is the exponential growth rate of the coefficient sum in degrees at most $k d_G/3$ of $H_G(t)^k$ as $k\to\infty$

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds $U_n(\mathbb F_p)$ for primes $p\leq7$, $3\leq n\leq6$ and $(p-1)n(n^2-1)/6\leq80$, together with $\operatorname{Heis}_r(\mathbb F_p)$ for the same primes and $2\leq r\leq4$; the rank-one Heisenberg cases are the $U_3$ rows. These 25 groups cover small characteristics, varying nilpotency class and varying generator count at class two. The prime, size, rank and degree bounds are noncanonical reference choices that keep the longest expanded polynomial to 1007 characters)

How they were obtained:

Computed by exact multiplication in $\mathbb Z[t]$. An independent logarithmic-derivative coefficient recurrence agrees on all 25 entries. Degree, constant and leading coefficients, coefficient symmetry, first-layer dimension, group order and the derivative at $t=1$ are checked for every row.

Direct linear algebra in the group algebras checks the complete augmentation filtration for $U_3(\mathbb F_p)$ with $p=2,3,5,7$, for $U_4(\mathbb F_2)$ and $U_5(\mathbb F_2)$, and for $\operatorname{Heis}_r(\mathbb F_2)$ with $r=2,3,4$. The direct method first passes cyclic groups of orders $2$, $3$ and $5$ as controls. The checks also compare the $B_2$ polynomial and all 13 relevant type $A$ order polynomials with the existing companion tables. The attached generator, checks and argument-selection note describe the computation and its coverage.