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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 18th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI see there's no page for invariant theory. Is there a slick nPOV on what it is all about? I see [here](http://mathoverflow.net/questions/166/resources-on-invariant-theory) Ben Webster explaining about invariant theory and geometric invariant theory.

   I see there’s no page for invariant theory. Is there a slick nPOV on what it is all about? I see here Ben Webster explaining about invariant theory and geometric invariant theory.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 18th 2012
   • (edited Jan 18th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexBelow that link the [reply by Andrea Ferretti](http://mathoverflow.net/questions/166/resources-on-invariant-theory/16601#16601) looks good to me: - invariant theory studies algebraic objects (elements of rings) invariant under group actions; - geometric invariant theory interprets this under [[Isbell duality|algebra/geometry duality]] as the study of quotient spaces. I have created brief stubs _[[invariant theory]]_ and _[[geometric invariant theory]]_ .

   Below that link the reply by Andrea Ferretti looks good to me:

   • invariant theory studies algebraic objects (elements of rings) invariant under group actions;

   • geometric invariant theory interprets this under algebra/geometry duality as the study of quotient spaces.

   I have created brief stubs invariant theory and geometric invariant theory .

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 18th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI was wondering about the relationship to representation theory from p. 48 of [this](http://www1.maths.leeds.ac.uk/~pmtwc/repinv.pdf). So if $$ \rho: B G \to Vect_{\mathbb{C}} $$ is a representation, with $U$ the vector space, it seems to be that invariants are polynomial functions $f: U \to \mathbb{C}$, such that $f(g u) = f(u)$. Does that mean they are intertwiners between a representation and the trivial one-dimensional rep?

   I was wondering about the relationship to representation theory from p. 48 of this. So if

   $$\rho: B G \to Vect_{\mathbb{C}}$$

   is a representation, with $U$ the vector space, it seems to be that invariants are polynomial functions $f: U \to \mathbb{C}$, such that $f(g u) = f(u)$. Does that mean they are intertwiners between a representation and the trivial one-dimensional rep?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 18th 2012
   • (edited Jan 18th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi David, yes, if you take the space of polynomials to be the representation (I mean, in your comment you'd need to separate the symbol $U$ into two different symbols). So if $\rho : \mathbf{B}G \to Vect$ is a representation on a vector space $V$, and $k$ is the tensor unit representation, then $Hom(k, \rho) \subset V$ is the subset of invariant vectors in $V$ (fixed points). More generally, for $G$ an $\infty$-group acting on an object $X$ in some $\infty$-topos, with the action exhibited by the action $\infty$-groupoid fiber sequence $$ X \to X //G \stackrel{p_\rho}{\to} \mathbf{B}G $$ the invariants _in_ $X$ are the sections of $p_\rho$. If instead you want the invariants _on_ $X$ with coefficients in some $A$, then you want the invariants _in_ the internal hom $[X,A]$ with respect to the canonically induced action.

   Hi David,

   yes, if you take the space of polynomials to be the representation (I mean, in your comment you’d need to separate the symbol $U$ into two different symbols).

   So if $\rho : \mathbf{B}G \to Vect$ is a representation on a vector space $V$, and $k$ is the tensor unit representation, then $Hom(k, \rho) \subset V$ is the subset of invariant vectors in $V$ (fixed points).

   More generally, for $G$ an $\infty$-group acting on an object $X$ in some $\infty$-topos, with the action exhibited by the action $\infty$-groupoid fiber sequence

   $$X \to X //G \stackrel{p_\rho}{\to} \mathbf{B}G$$

   the invariants in $X$ are the sections of $p_\rho$.

   If instead you want the invariants on $X$ with coefficients in some $A$, then you want the invariants in the internal hom $[X,A]$ with respect to the canonically induced action.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 18th 2012
   • (edited Jan 18th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex...where the last sentence brings us to _geometric_ invariant theory: we we observe that, under good conditions, the invariant functions in $[X,A]$ are the functions $[X//G, A]$ on the quotient space. (I should try to say that more precisely in the $\infty$-topos context....)

   …where the last sentence brings us to geometric invariant theory:

   we we observe that, under good conditions, the invariant functions in $[X,A]$ are the functions $[X//G, A]$ on the quotient space.

   (I should try to say that more precisely in the $\infty$-topos context….)

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeJan 18th 2012
   • PermaLink
   Author: zskoda
   Format: MarkdownItexFor some reason in GIT it is often very important to construct coarse moduli spaces and not fine moduli stacks. Probably somebody will take this into account, I am not quite there.

   For some reason in GIT it is often very important to construct coarse moduli spaces and not fine moduli stacks. Probably somebody will take this into account, I am not quite there.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJan 18th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI wrote: > (I should try to say that more precisely in the ∞-topos context....) For the internal hom as I stated it one may need some $\sharp$-escape route. For the external hom, however, the relation between invariants and geometric invariants should be straigthforward. The quotient $X//G$ is the (homotopy) colimit over the $G$-action $$ X // G \simeq {\lim_\to}_{\mathbf{B}G} X $$ So functions on the quotients -- hence elements in $\mathbf{H}(X//G, A)$ -- form the limit $$ \mathbf{H}(X//G, A) \simeq \mathbf{H}( {\lim_\to}_{\mathbf{B}G} X , A) \simeq {\lim_\leftarrow}_{\mathbf{B}G} \mathbf{H}(X , A) \,, $$ which are the invariants, as before. I don't have the leisure now to flesh this out more in detail. But that's the general mechanism here, I think.

   I wrote:

   (I should try to say that more precisely in the ∞-topos context….)

   For the internal hom as I stated it one may need some $\sharp$-escape route.

   For the external hom, however, the relation between invariants and geometric invariants should be straigthforward.

   The quotient $X//G$ is the (homotopy) colimit over the $G$-action

   $$X // G \simeq {\lim_\to}_{\mathbf{B}G} X$$

   So functions on the quotients – hence elements in $\mathbf{H}(X//G, A)$ – form the limit

   $$\mathbf{H}(X//G, A) \simeq \mathbf{H}( {\lim_\to}_{\mathbf{B}G} X , A) \simeq {\lim_\leftarrow}_{\mathbf{B}G} \mathbf{H}(X , A) \,,$$

   which are the invariants, as before.

   I don’t have the leisure now to flesh this out more in detail. But that’s the general mechanism here, I think.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 19th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexHmm, so what's so 'geometric' about geometric invariant theory? If Todd and Jim are [talking](http://golem.ph.utexas.edu/category/2007/11/concrete_groups_and_axiomatic_1.html) about the action of $G$ on $P(x^n)$, well that's just the $G$-invariant functions in $[X^n, 2]$, which is $[X^n/G, 2] = P(X^n/G)$. Is that geometric? I guess the discussion did take place with regard to the [seminar](Geometric Representation Theory) on Geometric Representation Theory.

   Hmm, so what’s so ’geometric’ about geometric invariant theory? If Todd and Jim are talking about the action of $G$ on $P(x^n)$, well that’s just the $G$-invariant functions in $[X^n, 2]$, which is $[X^n/G, 2] = P(X^n/G)$. Is that geometric? I guess the discussion did take place with regard to the [seminar](Geometric Representation Theory) on Geometric Representation Theory.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJan 19th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Hmm, so what's so 'geometric' about geometric invariant theory? Traditionally it is considered in the context of algebraic geometry. There one is interested in function rings more interesting than the maps on discrete sets that appear in the example that you are looking at. Accordingly, their formal duals are more interesting. Also, in this context the general abstract duality often kicks one out of any given subclass of nice objects into the more general world of sheaves. In order to avoid or fight or ignore that, one needs to have suitable technical assumptions, and that's what makes up much of the theory. As far as I am aware, at least. I am not an expert on GIT.

   Hmm, so what’s so ’geometric’ about geometric invariant theory?

   Traditionally it is considered in the context of algebraic geometry. There one is interested in function rings more interesting than the maps on discrete sets that appear in the example that you are looking at. Accordingly, their formal duals are more interesting.

   Also, in this context the general abstract duality often kicks one out of any given subclass of nice objects into the more general world of sheaves. In order to avoid or fight or ignore that, one needs to have suitable technical assumptions, and that’s what makes up much of the theory.

   As far as I am aware, at least. I am not an expert on GIT.

10. • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeJan 19th 2012
    • (edited Jan 19th 2012)
    • PermaLink
    Author: zskoda
    Format: MarkdownItex> any given subclass of nice objects into the more general world of sheaves. In order to avoid or fight or ignore that I am not sure, but I think if you are interested (by the nature of classical problems in representation theory, theory of binary forms etc.) in some SET (or subring) of invariants than it does not help that there is in geometric picture some stacky replacement. So it is not a wish to avoid but a need to avoid. On the other hand, Mumford has the things sometimes backwards -- to _construct_ nice moduli spaces (what is a different problem) one uses tools of the type geometric invariant theory was doing for its original problems. So the construction of moduli spaces, has, by the akin technique and closeness being added to the original problem. Thus one can add higher categorical analogues into the consideration but one should also respect that some of the classical problems of invariant theory, do not profit (nor could be solved) by going into a derived replacement.

    any given subclass of nice objects into the more general world of sheaves. In order to avoid or fight or ignore that

    I am not sure, but I think if you are interested (by the nature of classical problems in representation theory, theory of binary forms etc.) in some SET (or subring) of invariants than it does not help that there is in geometric picture some stacky replacement. So it is not a wish to avoid but a need to avoid. On the other hand, Mumford has the things sometimes backwards – to construct nice moduli spaces (what is a different problem) one uses tools of the type geometric invariant theory was doing for its original problems. So the construction of moduli spaces, has, by the akin technique and closeness being added to the original problem. Thus one can add higher categorical analogues into the consideration but one should also respect that some of the classical problems of invariant theory, do not profit (nor could be solved) by going into a derived replacement.

11. • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 24th 2012
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI see [[invariant theory]] has [[invariant polynomial]] under 'examples'. Is there a simple way to see why it's an example?

    I see invariant theory has invariant polynomial under ’examples’. Is there a simple way to see why it’s an example?

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 24th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItexHi David, I have added a paragraph in reply [here](http://ncatlab.org/nlab/show/invariant+theory#Examples).

    Hi David,

    I have added a paragraph in reply here.

13. • CommentRowNumber13.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 25th 2012
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexHi Urs, Thanks for that. Is it worth including what you wrote at #4 and #7 on the page?

    Hi Urs,

    Thanks for that.

    Is it worth including what you wrote at #4 and #7 on the page?

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJan 25th 2012
    • PermaLink
    Author: Urs
    Format: MarkdownItex> Is it worth including what you wrote at #4 and #7 on the page? I was thinking about doing so. It's not something one sees discussed in texts that are titled "invariant theory", but I think it helps to see what's going on in the background. I am a bit busy now, but I'll try to add something along these lines a little later.

    Is it worth including what you wrote at #4 and #7 on the page?

    I was thinking about doing so. It’s not something one sees discussed in texts that are titled “invariant theory”, but I think it helps to see what’s going on in the background.

    I am a bit busy now, but I’ll try to add something along these lines a little later.

15. • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 20th 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexShouldn't invariant theory make for a happy hunting ground for HoTT-ers? If vector spaces aren't to your taste, there's also the [groupoidifcation](https://arxiv.org/abs/1710.10307) approach: >Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids. Of course, there's plenty of sophisticated push-pull span-ish stuff to be found about the nLab, such as at [[motivic quantization]], but what about basic classical invariant theory? Perhaps this is more the domain of computer algebra programs. Peter Olver wonders in his [book](http://www.cambridge.org/us/academic/subjects/mathematics/algebra/classical-invariant-theory) what Cayley might have achieved with such a system. Is there any idea that such programs might be integrated with Coq?

    Shouldn’t invariant theory make for a happy hunting ground for HoTT-ers? If vector spaces aren’t to your taste, there’s also the groupoidifcation approach:

    Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids.

    Of course, there’s plenty of sophisticated push-pull span-ish stuff to be found about the nLab, such as at motivic quantization, but what about basic classical invariant theory?

    Perhaps this is more the domain of computer algebra programs. Peter Olver wonders in his book what Cayley might have achieved with such a system. Is there any idea that such programs might be integrated with Coq?

16. • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeAug 20th 2018
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexWell, the abstract versions of #4 and #7 make perfect sense in HoTT.

    Well, the abstract versions of #4 and #7 make perfect sense in HoTT.

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 20th 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexYes, but has anyone considered the kind of results that might be in range?

    Yes, but has anyone considered the kind of results that might be in range?

18. • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeAug 20th 2018
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexNot that I know of. The closest work I know of is [Higher Groups in Homotopy Type Theory](https://arxiv.org/abs/1802.04315).

    Not that I know of. The closest work I know of is Higher Groups in Homotopy Type Theory.

19. • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 22nd 2018
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexRe computer algebra (#15), I see people are thinking about how it could work with Coq, [here](http://www-ljk.imag.fr/membres/Dominique.Duval/CACOS12/LoicPottier.pdf).

    Re computer algebra (#15), I see people are thinking about how it could work with Coq, here.