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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 30th 2018
   • (edited Sep 30th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexam starting some minimum here, am really trying to see what is known regarding the following: Since, by the McKay correspondence, we may identify each vertex of a Dynkin quiver with the isomorphism class of an irreducible representation of the corresponding [[finite subgroup of SU(2)]] $G_{ADE} \subset SU(2)$, a [[Bridgeland stability condition]] on representations of a Dynkin quiver directly restricts to a stability function on $G_{ADE} Rep$. But it feels that stability functions on the representation ring $$ R_{\mathbb{C}}(G_{ADE}) = K\left( G_{ADE} Rep\right) \longrightarrow \mathbb{C} $$ ought to have a really elementary expression in terms of basic objects of representation theory. Can one say anything here? In particular, the immediate reaction when asked to present a complex-valued function on reps is to just use their characters, maybe evaluated at some chosen conjugacy class, and probably normalized in some way. Is this known? Are there at least examples of stability functions on $G_{ADE}$-representation which have an elementary representation-theoretic expression, hopefully in terms of characters? This seems like it should almost be the first non-trivial example of stability conditions, but I have trouble finding any source that would make this explicit. <a href="https://ncatlab.org/nlab/revision/Dynkin+quiver/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Dynkin+quiver">current</a>

   am starting some minimum here, am really trying to see what is known regarding the following:

   Since, by the McKay correspondence, we may identify each vertex of a Dynkin quiver with the isomorphism class of an irreducible representation of the corresponding finite subgroup of SU(2) $G_{ADE} \subset SU(2)$, a Bridgeland stability condition on representations of a Dynkin quiver directly restricts to a stability function on $G_{ADE} Rep$.

   But it feels that stability functions on the representation ring

   $$R_{\mathbb{C}}(G_{ADE}) = K\left( G_{ADE} Rep\right) \longrightarrow \mathbb{C}$$

   ought to have a really elementary expression in terms of basic objects of representation theory. Can one say anything here?

   In particular, the immediate reaction when asked to present a complex-valued function on reps is to just use their characters, maybe evaluated at some chosen conjugacy class, and probably normalized in some way.

   Is this known? Are there at least examples of stability functions on $G_{ADE}$-representation which have an elementary representation-theoretic expression, hopefully in terms of characters?

   This seems like it should almost be the first non-trivial example of stability conditions, but I have trouble finding any source that would make this explicit.

   v1, current