A correction to my attempted correction in #14: I’d written

When $G$ is finite, the ring $A = \mathbf{C}[G]$ is semisimple, and this implies that for any homomorphism $\theta: K_0(A\text{-}\mathbf{mod}) \to \Z$, every $A$-module $E$ is $\theta$-polystable (direct sum of $\theta$-stables); it is $\theta$-stable if and only if it is simple, that is, irreducible as a representation of $G$.

What I meant was that an $A$-module $E$ is $\theta$-semistable if and only if it is $\theta$-polystable, that is, isomorphic to a direct sum of $\theta$-stable $A$-modules; it is $\theta$-stable if and only if $\theta(E) = 0$ and $E$ is simple, that is, irreducible as a representation of $G$.

Incidentally, the following fundamental work about stability structures on abelian categories may be of interest here:

- A. Rudakov, Stability for an abelian category, J. Algebra
**197**(1997), 231–245.

Rudakov wrote some sequels to this article, but I haven’t read them.

]]>Thanks, your comment were very helpful. It was good for me to look into some basics here.

Luckily, today I realized that for the application that I have in mind I may not need stability conditions after all.

]]>Your analysis is correct, and I was hasty when I said in #11 that I expect that varying the homomorphism $\theta$ can produce a different set of $\theta$-stable representations.

When $G$ is finite, the ring $A = \mathbf{C}[G]$ is semisimple, and this implies that for any homomorphism $\theta: K_0(A\text{-}\mathbf{mod}) \to \Z$, every $A$-module $E$ is $\theta$-polystable (direct sum of $\theta$-stables); it is $\theta$-stable if and only if it is simple, that is, irreducible as a representation of $G$. So the stability structures of King don’t give anything different for $A$ from I’d mentioned in #6.

My apologies in case my comment in #11 led you astray.

]]>Okay, if the simple statement of Def. 1.1 with Thm. 4.1 in King 94 gives stability conditions on $G Rep_{\mathbb{C}}$ compatible with the “modern” meaning as used by Bridgeland etc., then that’s exactly the kind of simplification I was hoping for, in the case of a hands-on abelian category like $G Rep$.

So, as you said above, a stability condition on $G Rep_{\mathbb{C}}$ should just be a linear function

$R_{\mathbb{C}}(G) = K(G Rep_{\mathbb{C}}) \overset{ \phantom{AA} \theta \phantom{AA} }{\longrightarrow} \mathbb{Z}$from the represention ring to the integers,

and an element $V \in G Rep_{\mathbb{C}}$ is to be called stable with respect to $\theta$ if

$\theta(V) = 0$

$W \subset V$ with $W \neq V$ implies $\theta(W) \lt \theta(V)$.

(I am just repeating this for the record, and because I am taking the liberty of switching from King’s “$\gt$” to Bridgeland’s “$\lt$”.)

So let me explore that a bit.

In these terms, what is that “standard” stability condition which we talked about, the one coming from slope-stability of equivariant vector bundles, when restricting to equivariant vector bundles over the point. I suppose that standard stability condition should in fact be the one corresponding to $\theta$ vanishing identically

$\theta = 0 \,,$because that indeed makes the irreps and only the irreps be stable.

Does that sound right?

(Of course this is elementary, but I need to warm up to this stability business here.)

Okay, so what else can we get? In general, we can identify

$\theta = (\theta^i) \in \mathbb{Z}^{ Irreps(G) }$given by sending a rep

$V \;=\; \underset{i \in Irreps(G)}{\bigoplus} \underset{ \in \mathbb{Z} }{\underbrace{\langle V,V_i\rangle}} \, V_i$to

$\theta(V) \;\coloneqq\; \underset{i}{\sum} \theta^i \langle V,V_i\rangle \,.$Hm, so which other choice of $\theta$ would be interesting?

Hm. Is there any non-vanishing choice of $\theta$ that would have a non-trivial space of stable objects here?

Looks like as soon as $\theta^i$ is non-vanishing, then $V_i$ may not appear in a direct summand of a stable $V$. Because if $\theta^i$ is positive, then $V_i$ may not be a subobject of a stable object, while if it is negative, then it can only appear as a summand together with another object that contributes positively, but then *that* object may not appear as a subobject.

So then the only thing that can happen is that the stable objects are a subset of all irreps. That would be a bit sad.

Hm. I am missing something here.(?)

[edit: I think the issue is that King’s $\theta$ plays the role of the slope, but the slope itself is not meant to be additive, while King asks $\theta$ to be additive. That seems to be an overly strong condition.]

]]>Excellent, that’s just what I was looking for. I’ll have a look at that article by King, thanks again.

]]>Could it be that there are, in addition, other possible choices of stability conditions on $G$-representations, possibly with other stable objects?

I regret that I have not read about Bridgeland stability conditions or about the McKay correspondence, so I cannot say anything about most of what you have written. However, even from my limited perspective, I think that the above should be true. For example, if $G$ is finite, $A = \mathbf{C}[G]$, and $\mathcal{A} = A\text{-}\mathbf{mod}$, then every homomorphism $\theta: K_0(\mathcal{A}) \to \mathbf{Z}$ defines a stability structure on the abelian category $\mathcal{A}$, as in Definition 1.1 of

- A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford 45 (1994), 515–530.

The moduli space $M(\theta)$ of $\theta$-semistable objects of $\mathcal{A}$ may vary as $\theta$ varies. In the cases that I’ve seen, where instead of $\mathbf{C}[G]$, I take $A$ to be the path algebra of a finite quiver, and consider $\theta$ real-valued instead of integer-valued, then the space of all $\theta$ is a finite-dimensional real affine space, with a distinguished hyperplane arrangement $\mathcal{H}$. As $\theta$ varies within a facet of $\mathcal{H}$, the moduli spaces $M(\theta)$ lie in the same isomorphism class. When $\theta$ crosses a wall of $\mathcal{H}$, the isomorphsim class of $M(\theta)$ may change. (This kind of phenomenon is described even as early as in Mehta and Seshadri’s work on parabolic vector bundles over a curve. The wall-crossing phenomenon is explicitly described in some situations by Thaddeus and others in terms of flips.)

]]>Okay, thanks.

Could it be that there are, in addition, other possible choices of stability conditions on $G$-representations, possibly with other stable objects?

I am thinking of the following, but please set me straight if this seems wrong:

For $G \subset SU(2)$ a finite subgroup of $SU(2)$, the K-theoretic McKay correspondence (GSV83) gives an isomorphism

$R_{\mathbb{C}}(G) \simeq K_G(\ast) \overset{\simeq}{\to} K(\tilde X)$from the representation ring to the $K$-theory over the resolution $\tilde X$ of the corresponding du Val singularity.

Now there is an interesting space of stability conditions over $\tilde X$, due to Bridgeland 05, hence of stability functions

$K(\tilde X) \overset{Z}{\longrightarrow} \mathbb{C}$But these pull back along the above isomorphism to stability functions

$R_{\mathbb{C}}(G) \overset{Z}{\longrightarrow} \mathbb{C}$on the representation ring, hence on the K-theory of equivariant vector bundles over the point.

Maybe the class of stable $G$-representations can change as we move around in that space of stability conditions?

Possibly the “standard” stability condition that you point out, with the irreps being precisely the stable objects, is “just” one point in this space of stability conditions?

I suppose page 2 in Bridgeland 05 is sort of about this. But I am hoping that after pullback to $R_{\mathbb{C}}(G)$ the stability functions $Z$ can be given a simple explicit description in terms of characters.

]]>regarding your pointer to Huybrechts-Lehn 96: Their Def. 1.2.3 seems to to be stated a in the non-equivariant context, no?

The $G$-semistability of a $G$-sheaf is usually defined to be just the semistability of the underlying sheaf. Thus, a $G$-sheaf $E$ is defined to be $G$-semistable if $F \preceq E$ (in the chosen preorder) for every non-zero proper subsheaf $F$ of $E$. See:

- Seshadri, page 32: “We say that a π-vector bundle V on X is π-semi-stable (or semi-stable π-bundle) if the underlying vector bundle of V is semi- stable.”

On the other hand, the $G$-stability of a $G$-sheaf $E$ is defined by requiring that $F \prec E$ (in the chosen preorder) for every non-zero proper $G$-subsheaf $F$ of $E$. See:

Seshadri, page 32: “A π-vector bundle V on X is said to be π-stable if V is π-semi-stable and for every proper π-sub-bundle W of V , we have µ(W ) < µ(V ).” (The condition of $\pi$-semistability, that is semistability of the underlying sheaf, need not be assumed, because it follows from the other condition, see Garcia-Prada, Proposition 7.)

Garcia-Prada, page 533: “Definition: The bundle $E$ is $G$-invarίantly stable with respect to $\omega$ if for every $G$-ίnvarίant coherent subsheaf $F$ with $0 < \rk(E) < \rk(F)$, we have $\mu(F) < \mu(E)$.”

Are you saying that (semi-)stability of equivariant coherent sheaves should be just that of their underlying plain coherent sheaves?

As per the above definitions, the action of $G$ does not affect the definition of the $G$-semistability of a $G$-sheaf. It does enter the definition of $G$-stability of a $G$-sheaf through the condition that the strict preorder relation holds only for $G$-subsheaves and not necssarily for all subsheaves. Also, the action of $G$ does not affect the definition of the preorder in the situations that I’ve seen; e.g., in both the above references the slope $\mu$ is the slope of the underlying sheaf, and does not take into account the action of $G$.

Why would that be so?

I don’t know. However, it works, where I can specify the meaning of “works” to some extent.

]]>Allow me to ask regarding your pointer to Huybrechts-Lehn 96:

Their Def. 1.2.3 seems to to be stated a in the non-equivariant context, no? They seem to introduce group actions not before section 4.2.

Are you saying that (semi-)stability of equivariant coherent sheaves should be just that of their underlying plain coherent sheaves? Why would that be so?

(I am not doubting it, just trying to understand.)

]]>Thanks a million! I’ll have a look.

]]>Supposedly there is a generalization to stable equivariant vector bundles? … Is there any literature on this?

Here are a couple of references:

C. S. Seshadri, Moduli of π-vector bundles over an algebraic curve. 1970. Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) pp. 139–260. Edizioni Cremonese, Rome. LaTeX version. Constructs the moduli space of (semi)stable $\pi$-vector bundles over a compact Riemann surface $X$, where $\pi$ is a finite group acting on $X$.

O. García-Prada, Invariant connections and vortices. Comm. Math. Phys. 156 (1993), no. 3, 527–546. Considers stability of $G$-vector bundles on a compact K"ahler manifold $X$, where $G$ is a compact Lie group acting on $X$.

If so, then in the special case over the point there should be an answer to: When is a G-representation stable, in this sense?

I guess that, according to any of the usual definitions of $G$-stability, every $G$-vector bundle on a point $*$ is semistable. It is stable if and only if it is irreducible as a representation of $G$. For instance, in the notation of [Huybrechts and Lehn, Definition 1.2.3], $p(E,m)=1$ for every non-zero sheaf $E$ on $*$.

]]>Supposedly there is a generalization to stable *equivariant* vector bundles? If so, then in the special case over the point there should be an answer to: When is a G-representation stable, in this sense?

Is there any literature on this?

]]>this is provided by the Narasimhan–Seshadri theorem which establishes that the moduli space of flat connections on a Riemann surface is naturally a complex manifolds.

My understanding is that this is an application of the variant of the general picture of stable objecs from Mumford’s geometric invariant theory, which he designed precisely for moduli spaces). So yes, it fits with the general picture I wanted to (eventually become able to) spell out.

]]>I hope you are in phase with my thoughts here.

I am currently focusing on aspects of the geometric quantization of CS theory and the appearance of stable vector bundles there.

]]>We already have the entry stable bundle, which I created in Spring 2010. It seems it is essentially about the same notion.

Note also the entry stability where one of the points is

in geometry, in the construction of moduli spaces there is a notion of stable and semistable objects (e.g.

Mumford stabilityin geometric invariant theory); they can be taken with respect to some stability data; more recently those are important in physical applications and abstract versions appeared in the context of triangulated categories, most notably Bridgeland stability conditions.

I had an ambitious plan to create and integrate a circle of entries on that kind of stability (e.g. some pointers like Kontsevich’s stable curves and the related moduli space, used e.g. in CFT, Mumford stability, Bridgeland stability, stable bundle etc.). I hope you are in phase with my thoughts here.

]]>started *stable vector bundle*, but still vague