Flag Hilbert–Poincaré series and Igusa zeta functions of hyperplane arrangements
Joshua Maglione
Bielefeld University
math.uni-bielefeld.de/~jmaglione
www.slides.com/joshmaglione/cirm2021
Joint with ...
Igusa's zeta functions
Count roots of f∈Z[X1,…,Xd]: let p be a prime,
Nm(p)=#{X∈(Z/pmZ)d f(X)≡0modpm}
N_m(p) = \#\left\{X\in(\Z/p^m\Z)^d ~\middle|~ f(X)\equiv 0\mod p^m\right\}
Pf(T)=m=0∑∞p−dmNm(p)Tm
\displaystyle P_f(T) = \sum_{m=0}^\infty p^{-dm}N_m(p)T^m
Igusa's zeta function – p-adic integral closely related to
Of interest: f a product of linear polynomials.
Goal: use combinatorial & topological tools to understand asymptotics and arithmetic of Pf(T) and Igusa zeta function.
Comes up in computing: subgroup, representation, & class counting zeta functions.
Flag Hilbert–Poincaré series
A hyperplane arrangement – hyperplanes in Kd
L(A) intersection poset – all nonempty intersections
πA(Y)=x∈L(A)∑μ(x)(−Y)rk(x)=Poin(Cd∖H∈A⋃H;Y)
\pi_{\mathcal{A}}(Y) = \displaystyle\sum_{x\in\mathcal{L}(\mathcal{A})} \mu(x)(-Y)^{\mathrm{rk}(x)} = \mathrm{Poin}(\mathbb{C}^d \setminus \bigcup_{H\in\mathcal{A}}H;\; Y)
Poincaré polynomial:
K=C
1^
0^
Bn:={Xi−Xj ∣ 1≤i<j≤n}:
\mathcal{B}_n := \{X_i - X_j ~|~ 1 \leq i < j \leq n\}:
πB4(Y)=1+6Y+11Y2+6Y3
\pi_{\mathcal{B}_4}(Y) = 1 + 6Y + 11Y^2 + 6Y^3
L(B4)
rk
0
1
2
3
For F=(x1<⋯<xℓ)∈Δ(L(A)), with x0=0^, xℓ+1=∅,
πF(Y)=k=0∏ℓπAxk+1xk(Y)
\pi_F(Y) = \displaystyle\prod_{k=0}^{\ell} \pi_{\mathcal{A}^{x_k}_{x_{k+1}}}(Y)
With x∈L(A), two new arrangements
AxAx={H∈A ∣ x⊆H},={x∩H ∣ H∈A∖Ax,x∩H=∅}.(subarrangement)(restriction)
\begin{aligned}
\mathcal{A}_x &= \left\{H\in\mathcal{A} ~\middle|~ x\subseteq H \right\}, & & (\text{subarrangement})\\
\mathcal{A}^x &= \left\{x\cap H ~\middle|~ H\in\mathcal{A}\setminus\mathcal{A}_x,\; x\cap H\neq \varnothing \right\}. & & (\text{restriction})
\end{aligned}
Example: A=B4
Order complex: Δ(L(A)) – flags in L(A)
F empty flag in Δ(L(A)):
πF(Y)=πA∅0^(Y)=πA(Y)=1+6Y+11Y2+6Y3
\begin{aligned}
\pi_F(Y) &= \pi_{\mathcal{A}_{\varnothing}^{\hat{0}}}(Y) = \pi_{\mathcal{A}}(Y) \\
&= 1 + 6Y + 11Y^2 + 6Y^3
\end{aligned}
F=():
πF(Y)=πA0^(Y)⋅πA∅(Y)=(1+Y)⋅(1+3Y+2Y2)
\begin{aligned}
\pi_F(Y) &= \pi_{\mathcal{A}^{\hat{0}}}(Y) \cdot\pi_{\mathcal{A}_{\varnothing}}(Y) \\
&= (1+Y)\cdot (1 + 3Y + 2Y^2)
\end{aligned}
πF(Y)=πA0^(Y)⋅πA−(Y)⋅πA∅(Y)=(1+Y)⋅(1+Y)⋅(1+Y)
\begin{aligned}
\pi_F(Y) &= \pi_{\mathcal{A}^{\hat{0}}}(Y) \cdot \pi_{\mathcal{A}_{\phantom{-}}}(Y) \cdot \pi_{\mathcal{A}_{\varnothing}}(Y) \\
&= (1+Y)\cdot (1 + Y) \cdot (1 + Y)
\end{aligned}
F=(<):
Set L(A)=L(A)∖{0^} and A is central if ⋂H∈AH=∅
The flag Hilbert–Poincaré series:
fHPA(Y,T)=F∈Δ(L(A))∑πF(Y)x∈F∏1−TxTx
\mathsf{fHP}_{\mathcal{A}}(Y, \bm{T}) = \displaystyle\sum_{F\in\Delta(\widetilde{\mathcal{L}}(\mathcal{A}))} \pi_F(Y) \prod_{x\in F} \dfrac{T_x}{1 - T_x}
Thm (M.–Voll 2021). For A defined over characteristic 0 and central, then
fHPA(Y−1,(Tx−1)x∈L(A))=(−Y)−rk(A)T1^⋅fHPA(Y,T).
\mathsf{fHP}_{\mathcal{A}}\left(Y^{-1},(T_x^{-1})_{x\in\widetilde{\mathcal{L}}(\mathcal{A})}\right) = (-Y)^{-\mathrm{rk}(\mathcal{A})} T_{\hat{1}} \cdot \mathsf{fHP}_{\mathcal{A}}(Y, \bm{T}).
Self-reciprocity
Idea: fHPA is equivalent to a p-adic integral.
L(B4)
Igusa zeta function: a substitution of fHPA(Y,T).
YTx=−p−1,=⎩⎨⎧p−1−sp−2−2sp−2−3sp−3−6sx=−,x=−,x=−,x=−.
\begin{aligned}
Y &= -p^{-1}, \\
T_x &= \begin{cases}
p^{-1-s} & x = \phantom{-}, \\
p^{-2-2s} & x = \phantom{-}, \\
p^{-2-3s} & x = \phantom{-}, \\
p^{-3-6s} & x = \phantom{-}.
\end{cases}
\end{aligned}
(1−p−1−s)2(1−p−2−3s)(1−p−3−6s)(1−p−1)(1−5p−1+6p−2+⋯−6p−4−5s+5p−5−5s−p−6−5s)
\dfrac{(1-p^{-1})\left(1 - 5p^{-1} + 6p^{-2} + \cdots - 6p^{-4-5s} + 5p^{-5-5s} - p^{-6-5s}\right)}{(1 - p^{-1-s})^2(1 - p^{-2-3s})(1 - p^{-3-6s})}
Igusa zeta function of fB4: for all primes p,
fB4(X)=(X1−X2)(X1−X3)(X1−X4)(X2−X3)(X2−X4)(X3−X4)
f_{\mathcal{B}_4}(\bm{X}) = (X_1 - X_2)(X_1 - X_3)(X_1 - X_4)(X_2 - X_3)(X_2 - X_4)(X_3 - X_4)
Class counting zeta function: a substitution of fHPA(Y,T).
ζGcc(s)=m=0∑∞#cc(G(Z/pmZ))p−ms
\zeta^{\mathrm{cc}}_{\mathbf{G}}(s) = \displaystyle \sum_{m=0}^{\infty} \# \mathrm{cc}(\mathbf{G}(\mathbb{Z}/p^m\mathbb{Z})) p^{-ms}
G a nilpotent group scheme of finite type over Zp:
Set Γ=K3,2 and GΓ graphical group scheme.
GΓ(Zp) =
≤GL12(Zp)
=⟨I+E1,2+E5,7+E6,8, …,
I+E1,6+E2,8+E3,10+E4,12⟩
Cn coordinate hyperplanes in Kn and L(Cn)≅2[n].
YTI=−p−1,=⎩⎨⎧p6−∣I∣−sp7−∣I∣−sp8−∣I∣−sp4−sp−∣I∣I=−,I=−,I=−,I=−,I=−.
\begin{aligned}
Y &= -p^{-1}, \\
T_I &= \begin{cases}
p^{6-|I|-s} & I = \phantom{-}, \\
p^{7-|I|-s} & I = \phantom{-}, \\
p^{8-|I|-s} & I = \phantom{-}, \\
p^{4-s} & I = \phantom{-}, \\
p^{-|I|} & I = \phantom{-}.
\end{cases}
\end{aligned}
Class counting zeta function for GΓ(Zp) for all primes p:
(1−p3−s)(1−p5−s)21+p−s−p1−s−2p2−s−p3−s+p4−s+p4−2s
\dfrac{1 + p^{-s} - p^{1-s} - 2p^{2-s} - p^{3-s} + p^{4-s} + p^{4-2s}}{(1-p^{3-s})(1-p^{5-s})^2}
Details: Rossmann–Voll
A different coarsening
Set each Tx=T to get coarse flag Hilbert–Poincaré series:
Easier to see examples & other combinatorial properties.
Nice form:
cfHPA(Y,T)=(1−T)rk(A)NA(Y,T)
\mathsf{cfHP}_{\mathcal{A}}(Y, T) = \dfrac{\mathcal{N}_{\mathcal{A}}(Y, T)}{(1 - T)^{\mathrm{rk(\mathcal{A})}}}
cfHPA(Y,T)=F∈Δ(L(A))∑πF(Y)(1−TT)∣F∣
\mathsf{cfHP}_{\mathcal{A}}(Y, T) = \displaystyle\sum_{F\in\Delta(\widetilde{\mathcal{L}}(\mathcal{A}))} \pi_F(Y) \left(\dfrac{T}{1 - T}\right)^{|F|}
Braid arrangement in Qn:
Bn={Xi−Xj ∣ 1≤i<j≤n}
\mathcal{B}_n = \{X_i - X_j ~|~ 1\leq i < j \leq n\}
NA(Y,T)=F∈Δ(L(A))∑πF(Y)T∣F∣(1−T)rk(A)−∣F∣
\mathcal{N}_{\mathcal{A}}(Y, T) = \displaystyle\sum_{F\in\Delta(\widetilde{\mathcal{L}}(\mathcal{A}))} \pi_F(Y) T^{|F|}(1 - T)^{\mathrm{rk}(\mathcal{A})-|F|}
NB2(Y,T)=1+Y
\mathcal{N}_{\mathcal{B}_2}(Y, T) = 1 + Y
NB3(Y,T)=1+3Y+2Y2+(2+3Y+Y2)T
\mathcal{N}_{\mathcal{B}_3}(Y, T) = 1 + 3Y + 2Y^2 + (2 + 3Y + Y^2)T
NB4(Y,T)=1+6Y+11Y2+6Y3+(11+37Y+37Y2+11Y3)T+(6+11Y+6Y2+Y3)T2
\begin{aligned}
\mathcal{N}_{\mathcal{B}_4}(Y, T) &= 1 + 6Y + 11Y^2 + 6Y^3 \\
&\quad + (11 + 37Y + 37Y^2 + 11Y^3)T \\
&\quad + (6 + 11Y + 6Y^2 + Y^3)T^2
\end{aligned}
All nonnegative coefficients. Computed with HypIgu.
Do coefficients of NA(Y,T) have combinatorial interpretation?
Coxeter arrangements
The Eulerian polynomial En(T) is combinatorially defined.
E1:E2:E3:E4:E5:11126111146611112611
\begin{array}{rcccccccccc}
E_1: &&&&&& 1 &&&& \\
E_2: &&&&& 1 && 1 &&& \\
E_3: &&&& 1 && 4 && 1 && \\
E_4: &&& 1 && 11 && 11 && 1 & \\
E_5: && 1 && 26 && 66 && 26 && 1
\end{array}
NA(1,T)=πA(1)⋅Erk(A)(T).
\mathcal{N}_{\mathcal{A}}(1, T) = \pi_{\mathcal{A}}(1) \cdot E_{\mathrm{rk}(\mathcal{A})}(T).
Thm (M.–Voll 2021). For a Coxeter arrangement A with no E8 factors:
Conjecture (M.–Voll). For arbitrary A, the coefficients of NA(Y,T) are nonnegative.
Apply combinatorial & topological tools from hyperplane arrangements to understand asymptotics and arithmetic of
Pf(T)=m=0∑∞p−dmNm(p)Tm.
\displaystyle P_f(T) = \sum_{m=0}^\infty p^{-dm}N_m(p)T^m.
Flag Hilbert–Poincaré series have combinatorial features:
Thm. A central and characteristic 0 ⟹ fHPA self-reciprocal.
Thm. A Coxeter and no E8-factor ⟹ NA(1,T)=πA(1)⋅Erk(A)(T).
fHPA
Igusa
cc z.f.
cfHPA
Flag Hilbert–Poincaré series and Igusa zeta functions of hyperplane arrangements Joshua Maglione Bielefeld University math.uni-bielefeld.de/~jmaglione www.slides.com/joshmaglione/ cirm2021 Creative Commons Attribution 4.0 International License
Flag Hilbert--Poincaré series and Igusa zeta functions of hyperplane arrangements
By Josh Maglione
Flag Hilbert--Poincaré series and Igusa zeta functions of hyperplane arrangements
We define a class of multivariate rational functions associated with hyperplane arrangements called flag Hilbert--Poincaré series, and we show how these rational functions are connected to Igusa zeta functions. We report on a general self-reciprocity result and explore other connections within algebraic combinatorics. This is joint work with Christopher Voll.
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