Compute the Kac polynomial of a quiver

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Compute the Kac polynomial of a quiver #15187

Open 12d81aa2-5825-4c50-9de3-ede0c21017ed opened 10 years ago

12d81aa2-5825-4c50-9de3-ede0c21017ed commented 10 years ago

Compute the number of isomorphism classes of absolutely indecomposable representations of a quiver with a given dimension vector using the Kac-Stanley formula (see page 90 of Root systems, representations of quivers and invariant theory by Victor G. Kac).

Component: combinatorics

Keywords: quiver, representation, Kac-polynomial

Work Issues: add and test examples

Author: Amritanshu Prasad

Branch/Commit: public/15187 @ bcab72f

Issue created by migration from https://trac.sagemath.org/ticket/15187

tscrim commented 10 years ago

comment:1

Hey Amri,

Here's my initial work, but something isn't correct. I think what's going wrong is when I'm trying to do the log portion... I'm still looking into it.

Best,

Travis

New commits:

`45ad857`	`Merge branch 'develop' into u/tscrim/kac_polynomial`
`3d2096d`	`Merge branch 'u/tscrim/kac_polynomial' of trac.sagemath.org:sage into u/tscrim/kac_polynomial`
`8ec237d`	`Started Kac polynomials and DT-invariants.`
`e0362d5`	`Merge branch 'master' into u/tscrim/kac_polynomial`
`e3a32fc`	`Merge branch 'master' into u/tscrim/kac_polynomial`
`895db3a`	`Started Kac polynomials and DT-invariants.`

tscrim commented 10 years ago

Commit: 45ad857

tscrim commented 10 years ago

Branch: u/tscrim/kac_polynomial

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 10 years ago

Changed commit from 45ad857 to e2ac570

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 10 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

e2ac570 Merge branch 'develop' into u/tscrim/kac_polynomial

fchapoton commented 8 years ago

New commits:

cec5ca2 Merge branch 'u/tscrim/kac_polynomial' into 6.9.rc3

fchapoton commented 8 years ago

Changed branch from u/tscrim/kac_polynomial to public/15187

fchapoton commented 8 years ago

Changed commit from e2ac570 to cec5ca2

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from cec5ca2 to 5425bf8

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

`39996d6`	`Merge branch 'public/15187' into 6.10.beta2`
`5425bf8`	`trac #15187 cleanup, update, refactor, more doc`

fchapoton commented 8 years ago

Description changed:

--- 
+++ 
@@ -1 +1 @@
-Compute the number of isomorphism classes of absolutely indecomposable representations of a quiver with a given dimension vector using the Kac-Stanley formula (see *Root systems, representations of quivers and invariant theory* by Victor G. Kac).
+Compute the number of isomorphism classes of absolutely indecomposable representations of a quiver with a given dimension vector using the Kac-Stanley formula (see page 90 of *Root systems, representations of quivers and invariant theory* by Victor G. Kac).

fchapoton commented 8 years ago

Work Issues: add and test examples

fchapoton commented 8 years ago

Changed keywords from quivers, representations, Kac-polynomial to quiver, representation, Kac-polynomial

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Branch pushed to git repo; I updated commit sha1. New commits:

`731c2c3`	`Merge branch 'public/15187' into 7.1.b5`
`bcab72f`	`trac #15187 adding to the module list`

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 8 years ago

Changed commit from 5425bf8 to bcab72f

fchapoton commented 8 years ago

comment:11

I have made a new branch at u/chapoton/15187 where I try to use the correct formulas.

But I think that the naive approach is doomed to fail, as we probably need to compute inside the tensor product of r copies of Sym.

fchapoton commented 3 years ago

comment:12

see related stuff in #31950

fchapoton commented 2 years ago

comment:13

references here : https://ask.sagemath.org/question/47448/betti-numbers-of-nakajima-quiver-varieties/

and there : https://arxiv.org/abs/0811.1569