Is the F-polynomial of an indecomposable quiver representation irreducible ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:38:12Z http://mathoverflow.net/feeds/question/58648 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58648/is-the-f-polynomial-of-an-indecomposable-quiver-representation-irreducible Is the F-polynomial of an indecomposable quiver representation irreducible ? Kyungyong Lee 2011-03-16T15:25:43Z 2011-03-16T19:17:41Z <p>My question is as follows: is the F-polynomial of an indecomposable quiver representation irreducible as a polynomial, i.e. can it have a nontrivial factor? Here the F-polynomial is the generating function of the Euler characteristics of quiver Grassmannians, that is,</p> <p>F_M =\sum_{e_1,...e_n} \chi( Gr_{e_1,...e_n} M) x_1^e_1 ... x_n^e_n</p> <p>It is known that the F-polynomial of the direct sum of $M_1$ and $M_2$ is the product of $F_{M_1}$ and $F_{M_2}$.</p> http://mathoverflow.net/questions/58648/is-the-f-polynomial-of-an-indecomposable-quiver-representation-irreducible/58677#58677 Answer by David Speyer for Is the F-polynomial of an indecomposable quiver representation irreducible ? David Speyer 2011-03-16T19:17:41Z 2011-03-16T19:17:41Z <p>No.</p> <p>Consider the Kronecker quiver: two vertices with two arrows between them. Consider the representation with dimension vector $(3,3)$ given by the matrices <code>$$\begin{pmatrix} 1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \end{pmatrix}, \quad \begin{pmatrix} 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 0 &amp; 0 &amp; 0 \end{pmatrix}.$$</code></p> <p>The corresponding $F$-polynomial is computed in section 5 of <a href="http://front.math.ucdavis.edu/0604.5054" rel="nofollow">Caldero-Zelevinsky</a>, up to a monomial change of variables. (I won't try to get the monomial change of variables right.) It is <code>$$\begin{matrix} y^3 &amp; &amp; &amp; \\ + 3 y^2 &amp; + x y^2 &amp; &amp; \\ + 3y &amp; + 4 xy &amp; +x^2 y &amp; \\ +1 &amp; +3x &amp; +3x^2 &amp; +x^3 \end{matrix}$$</code></p> <p>This factors as <code>$$\begin{pmatrix} y &amp; \\ +1 &amp; + x \end{pmatrix} \begin{pmatrix} y^2 &amp; &amp; \\ +2y &amp; &amp; \\ +1 &amp; + 2x &amp; x^2 \end{pmatrix}.$$</code> </p> <p>Numerical experimentation reveals an interesting pattern: Let $F(d)$ be the $F$-polynomial of the indecomposable representation with dimension vector $(d,d)$. Then $F(n)$ appears to be prime if and only if $n+1$ is prime. Presumably, this should be an obvious consequence of the Chebyshev polynomial formulas in <a href="http://front.math.ucdavis.edu/0307.5082" rel="nofollow">Sherman-Zelevinsky</a>, but I don't see it right now.</p> <p>I can think of various ways you could try to salvage this, but I'll leave that project to you.</p>