Hi Nicolas. Thankyou for the reply. Sorry for my long reply...
I should clarify what my ultimate goal is here. I'm working on a
PhD. with Amnon Neeman.
I'm trying to refine the result in "An application of classical
invariant theory to identifiability in nonparametric mixtures" by
Elmore, Hall and Neeman. There, they work in the polynomial ring
$k[x_{ij], 1\leq i\leq r, 0\leq j\leq n$ with $k$ a field of
characteristic zero. The symmetric group $\Sigma_r$ on $r$ letters
permutes the $X_i=(x_{i0},\dotsc, x_{in})$. Thus for
$\sigma\in\Sigma_r$, $\sigma(X_i)=X_{\sigma(i)}$ (or if you prefer
from invariant theory $\sigma(X_i)=X_{\sigma^{-1}(i)}$; it doesn't
matter since they are interested in the invariant ring which is the
same either way).
They are interested in a certain set $M$ of $2^n$ invariant
polynomials (you can look at the paper for the specific
polynomials). The result they prove is that for each $r$, there is a
number $C(r)$ such that if $n\geq C(r)$, then the set $M$ generates
the field of rational invariants $k(x_{ij})^{\Sigma_r)$.
They give an estimate $c_1 \ln r \leq C(r) \leq c_2 r\ln r$. My task
is to find a better estimate for $C(r)$. In the paper, the authors
prove that work in the inverse limit ring (the limit $r\to\infty$) of
invariant polynomials $k[x_{ij}]^{\Sigma_r}$. Letting $S$ be the $k$
subalgebra generated by (limits of) symmetrized monomials, they show
$S$ is a polynomial ring in symmetrized monomials of the form
$X_1^{\alpha_1}$, $\alpha_1=(\alpha_{10},\dotsc,\alpha_{1n}) \in
\NN^{n+1}$. They also note that the surjective map $S\to
k[x_{ij}]^{\Sigma_r}$ has kernel spanned (over $k$) by symmetrized
monomials $X_1^{\alpha_1}\dotsm X_l^{\alpha_l}$ with $l>r$.
So finally, what I'm trying to do at the moment is investigate the
relations in $k[x_{ij}]^{\Sigma_r}$ between the symmetrized monomials
$X_1^{\alpha_1}$ or indeed any set of generators for
$k[x_{ij}]^{\Sigma_r}$. Thus the desire to work with multi-symmetric
functions rather than just polynomials.
I have some more comments below too (sorry!)
On Sat, Nov 20, 2010 at 09:05:58AM +0100, Nicolas M. Thiery wrote:
> Dear Paul, dear Emmmanuel Briand,
>
> On Sun, Nov 14, 2010 at 03:16:36PM -0800, Paul Bryan wrote:
> > I posted this question
> > http://ask.sagemath.org/question/200/multi-symmetric-functions-and-multi-partitions
> > on AskSage and it was suggested I post the question here. This is what
> > I originally asked:
> >
> > "Does sage support manipulating multi-symmetric functions/polynomials
> > and/or multi-partitions? Multi-symmetric functions are like the usual
> > symmetric ones, except the symmetric group acts by permuting "vectors"
> > of variables simultaneously, e.g. for an two vectors
> > $x=(x1,x2…),y=(y1,y2,…)$, $\Sigma_2$, acts by permuting $x,y$. A multi-
> > partition of a $n$-tuple $B=(b1,…,bn)$ of natural numbers is a
> > unordered set of $n$-tuples $A1,…,Al$ with $A1+⋯+Al=B$.
> >
> > I'd like to have a combinatorial class of multi-partitions with
> > similar functionality as partitions, e.g. .first(), .last() methods
> > and iter(). I'd also like to have a class like
> > SymmetricFunctionAlgebra, but with multi-symmetric functions instead.
> > I've had a bit of a poke around and there's some functionality in
> > Maxima (in the Sym) package that might help, but not quite like what I
> > want (that I can find). So, before writing code, I'm asking here.
> >
> > If the code needs to be written, I'm quite keen to make it my first
> > (hopefully of many) contribution to sage..."
>
> As far as I know, this is not in Sage yet, and it would be a nice
> feature to have.
>
> Nicolas Borie is doing his PhD on invariant rings of permutation
> groups and has patches in this direction in the queue. This includes
> multi-symmetric *polynomials* as a special case (i.e. when the vectors
> are of a fixed length m). Among other things, he has implemented the
> generation of all monomials of a given degree, up to the action of the
> permutation group, which gives a basis of the invariant ring. This
> should reduce to multi-partitions (of fixed length!), though I am
> unsure about a detail or two in your definition (your point is to get
> a basis of the multi-symmetric functions/polynomials, right?). The
> algorithmic might be far from optimal though in that special case. And
> it does not cover the infinite case.
It sounds like there is some overlap. Maybe I could get in touch with
Nicolas? I have almost finished coding an implementation of
multi-partitions including just what I need at the moment. I'll post
on sage-combinat-devel again when I've got something useful.
> In case you haven't done so, you probably want to check Emmanuel
> Briand's (in CC) publications who did his PhD on multi-symmetric
> polynomials:
>
> http://emmanuel.jean.briand.free.fr/publications/
>
I've read his paper "When is the Algebra of Multisymmetric Polynomials
generated by the Elementary Multisymmetric Polynomials ?".
> Emmanuel: would you have code to share to compute with them? Would you
> possibly be interested in joining a project to port it to Sage?
This would be great.
> > To give a little more info, I'm working on a problem trying to
> > determine when a certain set of polynomials generates the function
> > field of the multi-symmetric polynomials over a field.
>
> If I understand well the problem, I played a similar game a while ago
> with my favorite invariant ring (for a permutation group related to
> graphs). At the end of the day, a pinch of Galois theory did the
> trick: all I had to do was to take the usual elementary symmetric
> functions together with a single simple polynomial p (a primitive
> element for the extension field); the only requirement on p was that
> any permutation of the whole set of variables which left that
> polynomial p invariant had to be in the group. Maybe that could work
> for you as well. See http://arxiv.org/pdf/0812.3082. If French is ok
> for you, there are more details in my thesis:
> http://www.nicolas.thiery.name/Preprints/Thiery.IAGR/Thiery.IAGR.pdf
I'll take a look. I've been trying to figure out some way to use
Galois theory to solve my problem; it seems as though it might work,
but the snag is working with my particular generating set and the fact
that it is a generating set only for $r$ large enough. My initial
computation was to compute the degree of the field extension using the
algorithms described in "Gröbner bases applied to finitely generated
field extensions" by Muller-Quade and Steinwandt. This proved
intractable because of the Gröbner basis computation though the
symmetry of this problem should help. I haven't figure out how yet...
> Feel free to ask me further questions off-list if helpful!
I will!
> > As a first step, I'm trying to write some code to investigate the
> > relations between symmetrized monomials in the inverse limit ring,
> > i.e. the ring of multi-symmetric functions. Using Groebner bases
> > seems to be much too slow for this purpose. At the moment, I'm just
> > exploring this ring rather than trying to come up with an method of
> > solving the problem.
>
> Have fun!
>
> Best regards,
> Nicolas
> --
> Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
> http://Nicolas.Thiery.name/
>
> --
> You received this message because you are subscribed to the Google Groups
> "sage-combinat-devel" group.
> To post to this group, send email to sage-combinat-de...@googlegroups.com.
> To unsubscribe from this group, send email to
> sage-combinat-devel+unsubscr...@googlegroups.com.
> For more options, visit this group at
> http://groups.google.com/group/sage-combinat-devel?hl=en.
>
Thankyou very much if you read this far,
Paul.
--
You received this message because you are subscribed to the Google Groups
"sage-combinat-devel" group.
To post to this group, send email to sage-combinat-de...@googlegroups.com.
To unsubscribe from this group, send email to
sage-combinat-devel+unsubscr...@googlegroups.com.
For more options, visit this group at
http://groups.google.com/group/sage-combinat-devel?hl=en.
