ZZZZZZZZZZZZZZZZZZZ tttt
Z:::::::::::::::::Z ttt:::t
Z:::::::::::::::::Z t:::::t
Z:::ZZZZZZZZ:::::Z t:::::t
ZZZZZ Z:::::Z eeeeeeeeeeee ttttttt:::::ttttttt aaaaaaaaaaaaa
Z:::::Z ee::::::::::::ee t:::::::::::::::::t a::::::::::::a
Z:::::Z e::::::eeeee:::::et:::::::::::::::::t aaaaaaaaa:::::a
Z:::::Z e::::::e e:::::tttttt:::::::tttttt a::::a
Z:::::Z e:::::::eeeee::::::e t:::::t aaaaaaa:::::a
Z:::::Z e:::::::::::::::::e t:::::t aa::::::::::::a
Z:::::Z e::::::eeeeeeeeeee t:::::t a::::aaaa::::::a
ZZZ:::::Z ZZZZe:::::::e t:::::t ttttta::::a a:::::a
Z::::::ZZZZZZZZ:::e::::::::e t::::::tttt:::::a::::a a:::::a
Z:::::::::::::::::Ze::::::::eeeeeeee tt::::::::::::::a:::::aaaa::::::a
Z:::::::::::::::::Z ee:::::::::::::e tt:::::::::::tta::::::::::aa:::a
ZZZZZZZZZZZZZZZZZZZ eeeeeeeeeeeeee ttttttttttt aaaaaaaaaa aaaa
Version 0.2.1
by
Tobias Rossmann
Zeta provides methods for computing topological zeta functions arising from the enumeration of subalgebras, ideals, submodules, and representations of suitable algebraic structures. For theoretical background and descriptions of the methods used, see [1,2,3]. Zeta is distributed as a Python-package for the computer algebra system Sage.
This work is supported by the DFG Priority Programme “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory” (SPP 1489).
This software requires a recent version of Sage. It is untested on anything but 64-bit Linux (x86) systems.
In order to install Zeta, download
Zeta-0.2.1.tar.bz2 and extract it into some directory,
$DIR say. This will create a subdirectory
$DIR/Zeta.
Please make sure that any previous version of Zeta is either overwritten or removed beforehand.
If Sage is started from within
$DIR or if $DIR is among the directories
listed in $SAGE_PATH, then Zeta can be imported
via
import Zeta
If $DIR is not listed in $SAGE_PATH,
you can permanently include it by adding a line
export SAGE_PATH="$DIR:$SAGE_PATH"
to $HOME/.bashrc (if you use bash), where
$DIR is replaced by the actual directory name from
above; having applied this change, you will want to start a new
shell. Alternatively, for a temporary solution, after executing
sys.path.append('$DIR')
from within Sage, you should be able to import Zeta.
The following table lists all versions of Zeta released so far.
| File | Released |
|---|---|
| Zeta-0.2.1.tar.bz2 | 19/06/2015 |
| Zeta-0.2.tar.bz2 | 06/03/2015 |
| Zeta-0.1.tar.bz2 | 17/09/2014 |
You might also be interested in the list of algebras and associated topological zeta functions computed using Zeta.
If you receive a warning about Normaliz
being absent, it is strongly recommended that you install it. In
order to be found by Zeta, the executable file
normaliz should be either somewhere in your
$PATH or within $DIR/Zeta, where
$DIR is as above.
If you receive a warning about Zeta being unable to load a C extension, it is strongly recommended that you run
make clean ; make install
from within $DIR/Zeta.
Zeta will be fully functional but often much slower if Normaliz or the C extension is missing.
By an algebra, we mean a free $\mathbf Z$-module of finite rank endowed with a biadditive multiplication; we do not require this multiplication to be associative or Lie. Given a $\mathbf Z$-basis $x_1,\dotsc,x_d$ of an algebra $L$, define $\alpha_{ije}\in \mathbf Z$ by
\[ x_i x_j = \sum_{e=1}^d \alpha_{ije} x_e. \]The numbers $\alpha_{ije}$ are the structure constants of $L$ with respect to the chosen basis $(x_1,\dotsc,x_d)$. The principal method for specifying an algebra in Zeta is to provide structure constants as a nested list
\[ \begin{matrix} [[ (\alpha_{111},\dotsc,\alpha_{11d}), & \dotsc & (\alpha_{1d1},\dotsc,\alpha_{1dd}) ]\phantom], \\ \vdots & & \vdots \\ \phantom[[ (\alpha_{d11},\dotsc,\alpha_{d1d}), & \dotsc & (\alpha_{dd1},\dotsc,\alpha_{ddd}) ]] \\ \end{matrix} \]as the first argument of Zeta.Algebra. (We note
that the table of structure constants of an instance of
Zeta.Algebra is stored in the table
attribute.)
Given an algebra obtained via Zeta.Algebra, the
function Zeta.topological_zeta_function can be used to
attempt to compute an associated topological zeta function.
Specifically, Zeta.topological_zeta_function(L,
'subalgebras') will attempt to compute the topological subalgebra zeta
function of $L$ as a rational function in $s$, while
Zeta.topological_zeta_function(L, 'ideals') will do
the same for ideals.
If $L$ is a nilpotent Lie algebra, then
Zeta.topological_zeta_function(L, 'reps')
will attempt to compute the topological representation zeta function
of the unipotent algebraic group over $\mathbf Q$ corresponding to
$L\otimes_{\mathbf Z} \mathbf Q$.
In general, such computations are not guaranteed to succeed. If
the method for computing topological zeta functions from [1,2] (for subalgebras and ideals) or
[3] (for representations) fails,
Zeta.topological_zeta_function will raise an exception
of type Zeta.ReductionError. Disregarding bugs in Zeta,
Sage, or elsewhere, whenever Zeta.topological_zeta_function does
finish succesfully, its output is supposed to be correct.
To illustrate the computation of topological subobject zeta functions, consider the commutative algebra $L = \mathbf Z[X]/X^3$. As a $\mathbf Z$-basis of $L$, we choose $(1,x,x^2)$, where $x$ is the image of $X$ in $L$. The associated nested list of structure constants is
\[ \begin{matrix} [[(1, 0, 0), & (0, 1, 0), & (0, 0, 1)]\phantom],\\ \phantom[ [(0, 1, 0), & (0, 0, 1), & (0, 0, 0)]\phantom],\\ \phantom[[(0, 0, 1), & (0, 0, 0), & (0, 0, 0)]]. \end{matrix} \]The following documents a complete Sage session leading to the computation of the topological subalgebra and ideal zeta functions of $L$.
sage: import Zeta
sage: L = Zeta.Algebra([[(1, 0, 0), (0, 1, 0), (0, 0, 1)], [(0, 1, 0), (0, 0,1), (0, 0, 0)], [(0, 0, 1), (0, 0, 0), (0, 0, 0)]])
sage: Zeta.topological_zeta_function(L, 'subalgebras')
2*(15*s - 8)/((5*s - 4)*(3*s - 2)^2*s)
sage: Zeta.topological_zeta_function(L, 'ideals')
1/((3*s - 2)*(2*s - 1)*s)
We illustrate the computation of topological representation zeta functions of unipotent algebraic groups (over $\mathbf Q$) using the familiar example of the Heisenberg group $\mathbf H$. The first step is to construct a $\mathbf Z$-form of its Lie algebra. We choose the natural $\mathbf Z$-form $L = \mathbf Z x_1 \oplus \mathbf Z x_2 \oplus \mathbf Z x_3$ with $[x_1,x_2] = x_3$, $[x_2,x_1] = -x_3$ and $[x_i,x_j] = 0$ in the remaining cases. The list of structure constants of $L$ with respect to the basis $(x_1,x_2,x_3)$ is
\[ \begin{matrix} [[(0, 0, \phantom-0), & (0, 0, 1), & (0, 0, 0)]\phantom],\\ \phantom[ [(0, 0, -1), & (0, 0, 0), & (0, 0,0)]\phantom],\\ \phantom[[(0, 0, \phantom-0), & (0, 0, 0), & (0, 0, 0)]]. \end{matrix} \]The following documents a complete Sage session leading to the computation of the topological representation zeta function of $\mathbf H$.
sage: import Zeta
sage: L = Zeta.Algebra([[(0, 0, 0), (0, 0, 1), (0, 0, 0)], [(0, 0,-1), (0, 0, 0), (0, 0, 0)], [(0, 0, 0), (0, 0, 0), (0, 0, 0)]])
sage: Zeta.topological_zeta_function(L, 'reps')
s/(s - 1)
Zeta includes a “database” of algebras. When topological zeta functions associated with an algebra in the database have been succesfully computed using Zeta, these are stored as well.
Each algebra stored in Zeta can be referred to using its unique
identification number or one of finitely many names. Access to
these algebras is provided using the function
Zeta.lookup.
If Zeta.lookup is called with precisely one
argument entry, then entry should be
either an identification number or a name of an algebra, $L$ say,
in the database. In this case, Zeta.lookup will return
$L$. An optional second argument to Zeta.lookup can be
used to access other information about $L$:
'subalgebras',
'ideals', or 'reps',
then Zeta.lookup will return
the topological subalgebra, ideal, or representation zeta function of $L$
if it is known, and None otherwise.'id', then
Zeta.lookup returns the identification number of
$L$.'names', then
Zeta.lookup returns a list of the known names of
$L$.When called without arguments, Zeta.lookup returns
a list of pairs (i,names), where i ranges
over the identification numbers of all algebras in the database and
names is a possibly empty list of names associated
with the ith algebra.
The algebra $L = \mathbf Z[X]/X^3$ from Section 3.1 is known to Zeta under the name
'ZZ[X]/X^3'; it can be retrieved via L = Zeta.lookup('ZZ[X]/X^3').
We may recover the pre-computed topological zeta functions of $L$ as follows:
sage: Zeta.lookup('ZZ[X]/X^3', 'subalgebras')
2*(15*s - 8)/((5*s - 4)*(3*s - 2)^2*s)
sage: Zeta.lookup('ZZ[X]/X^3', 'ideals')
1/((3*s - 2)*(2*s - 1)*s)
Apart from self-explanatory names such as
'sl(2,ZZ)' and 'gl(2,ZZ)', Zeta also
includes algebras $L_{d,i}$, $L_{d,i}(\varepsilon)$, $L^i$,
$L^i_a$, $M^i$, and $M^i_a$ taken from de Graaf's tables of
nilpotent
and soluble Lie
algebras; their corresponding names in Zeta are of the form
'L(d,i)', 'L(d,i;eps)', 'L^i',
'L^i(a)', 'M^i', and 'M^i(a)'.
For the infinite families among these algebras, we only included
selected specialisations of the parameters.
Recall [1, Prop. 5.19(ii)] that the
topological subalgebra and ideal zeta functions of an algebra $L$
(over $\mathbf Z$) only depend on the $\mathbf C$-isomorphism type
of $L\otimes_{\mathbf Z}\mathbf C$;
a similar statement holds for topological representation zeta functions
by [3, Prop. 4.3].
Similar to Woodward's tables,
we use the notation 'g(...)' to refer to $\mathbf Z$-forms of
algebras from Seeley's list of
7-dimensional nilpotent Lie algebras over $\mathbf C$; for
example 'g(147A)' is a $\mathbf Z$-form of the algebra $1,4,7_A$
in Seeley's list.
The algebras 'N_i^(8,d)' are taken
from the lists of
Ren and Zhu,
and Yan and Deng.
The algebras called 'C(d,i)' and 'C(d,i;eps)' in
Zeta are “commutative versions” of the nilpotent Lie
rings 'L(d,i)' and 'L(d,i;eps)' respectively:
they were obtained by inverting the signs of all entries underneath
the diagonal in the matrices of structure constants.
An algebra called 'name[eps]' in Zeta is obtained by
tensoring 'name' with the dual numbers as
in [3, §6].
The function Zeta.examples.printall generates a
text-based list of
and writes these to an optional file-like object (which defaults
to stdout). The output of this function is also
available for download.
By the essential value of a rational function $Z\in
\mathbf Q(s)$ at a point $w\in \mathbf C$, we mean the value of
$Z/(s-w)^m$ at $s = w$, where $m$ is the order of $Z$ at $w$;
similarly, for $w = \infty$. The output of
Zeta.examples.printall (and hence the content of the
file linked to above) contains the essential values of topological
zeta functions at $0$ and $\infty$; these are related to
Conjectures IV–V from [1,2].
In [1,2], (topological) ideal zeta functions were treated as special cases of submodule zeta functions. In Zeta, we regard modules as special cases of algebras with operators. Namely, each algebra $L$ in Zeta is endowed with a possibly empty set $\Omega$ of operators, i.e. $\Omega$ consists of additive endomorphisms of $L$. The topological subalgebra and ideal zeta functions of $L$ are always understood to be those arising from the enumeration of $\Omega$-invariant subalgebras or ideals, respectively. Thus, if the multiplication of $L$ is trivial, then the $\Omega$-invariant subalgebras (and ideals) of $L$ are precisely the submodules of $L$ under the action of the enveloping associative ring of $\Omega$ within $\mathrm{End}(L)$.
In practice, $\Omega$ is given by a finite list of matrices (or
nested lists of integers representing those matrices) corresponding
to the defining basis of $L$. This list is then supplied to
Zeta.Algebra using the keyword parameter
operators. For algebras with zero multiplication,
instead of entering structure constants, you can provide a keyword
argument rank to Zeta.Algebra which
intialises all structure constants to zero.
We illustrate the computation of the topological submodule zeta function arising from the enumeration of sublattices within $\mathbf Z^3$ invariant under the matrix \[ \begin{bmatrix} 1 & 1 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \]
sage: M = Zeta.Algebra(rank=3, operators=[ [[1,1,-1],[0,1,1],[0,0,1]] ])
sage: Zeta.topological_zeta_function(M)
1/((3*s - 2)*(2*s - 1)*s)
In the database included with Zeta, for examples of algebras with trivial multiplication but non-empty lists of operators, we did not include ideal zeta functions; they coincide with the corresponding subalgebra and submodule zeta functions.
As an integral version of terminology used by Evseev, we say that
a $\mathbf Z$-basis $(x_1,\dotsc,x_d)$ of an algebra $L$ is
simple if each product $x_ix_j$ is of the form
$\varepsilon_{ij} x_{a_{ij}}$ for $\varepsilon_{ij} \in
\{-1,0,1\}$. In this case, the structure constants of $L$ with
respect to $(x_1,\dotsc,x_d)$ are determined by the matrix $A =
[\varepsilon_{ij} a_{ij}]_{i,j=1,\dotsc,d}$. Zeta supports the
creation of algebras from such a matrix $A$ by passing
simple_basis=True and matrix=$A$ as
arguments to Zeta.Algebra.
For example, the Heisenberg Lie ring with $\mathbf Z$-basis
$(x_1,x_2,x_3)$ and non-trivial products $[x_1,x_2] = x_3$ and
$[x_2,x_1] = -x_3$ from above can be defined in Zeta via
Zeta.Algebra(simple_basis=True, matrix=[[0,3,0], [-3,0,0],
[0,0,0] ]) .
(The following only applies to the computation of topological subalgebra
and ideal zeta functions.)
Computations using Zeta are usually very sensitive to the choice
of the basis used to define the structure constants of the algebra
under consideration. If a particular zeta function cannot be
directly computed using Zeta, it might be useful to consider
different bases. Given an algebra L of rank $d$ and an
invertible $d\times d$ matrix A over $\mathbf Z$, the
algebra obtained from $L$ by taking the rows of A as a
basis (relative to the original one) can be constructed via
L.change_basis(A).
If Zeta.topological_zeta_function is called with
the keyword argument optimise_basis=True, then Zeta
will attempt to find a basis of the algebra, $L$ say, in question
such that the associated toric datum is “small”.
Currently, Zeta simply loops over permutations of the defining
basis of $L$.
If Zeta.topological_zeta_function is called with
the keyword argument verbose=True, then detailed
information on the various stages of computations will be
displayed. Apart from illustrating the key steps explained in
[1,2,3], this
can often be helpful when it comes to estimating the feasibility of
the intended computation.
An upper bound on the number of CPUs used by
Zeta.topological_zeta_function can be enforced by
providing a numerical value for the keyword parameter
ncpus.
During computations of topological zeta functions, Zeta uses various temporary files. Be warned that for some computations carried out by the author, the combined size of these files exceeded 50 GB.
Zeta can also be quite demanding when it comes to system memory.
If computations run out of memory, you can try reducing the number
of CPUs used as indicated above or try setting the keyword
parameter profile of
Zeta.topological_zeta_function to
Zeta.Profile.SAVE_MEMORY. Setting
profile=Zeta.Profile.SPEED will result in slightly
better performance at the cost of increased memory use.
(The following only applies to the computation of topological subalgebra
and ideal zeta functions.)
The reduction step explained in [2]
depends on a strategy for chosing “reduction
candidates”. The default choice of a strategy is the one
explained in the paper. Alternative choices can be made using the
keyword parameter strategy of
Zeta.topological_zeta_function. In particular, setting
strategy=Zeta.Strategy.NONE disables reduction
completely, thus turning
Zeta.topological_zeta_function into an implementation
of [1] only. Passing
strategy=Zeta.Strategy.PREEMPTIVE will result in a
more aggressive reduction strategy which tries to anticipate and
remove causes of singularity in advance. While often slower than
the default, this strategy is needed to reproduce some of the
computations recorded in the database (Section 3.3).
[1] T. Rossmann. Computing topological zeta functions of groups, algebras, and modules, I. Proc. Lond. Math. Soc. (3) 110 (2015), no. 5, 1099–1134. (PDF)
[2] T. Rossmann. Computing topological zeta functions of groups, algebras, and modules, II. (preprint)
[3] T. Rossmann. Topological representation zeta functions of unipotent groups. (preprint)
Copyright © 2014, 2015 Tobias Rossmann.
Zeta is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
Zeta is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with Zeta. If not, see http://www.gnu.org/licenses.
Tobias Rossmann
Fakultät für Mathematik
Universität Bielefeld
Postfach 100131
D-33501 Bielefeld
Germany
tobias.rossmann (at) googlemail.com