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Irreducible components of varieties of representations I. The local case | Birge Huisgen-Zimmermann
; | Date: |
13 Oct 2014 | Abstract: | Let $Lambda$ be a local truncated path algebra over an algebraically closed
field $K$, i.e., $Lambda$ is a quotient of a path algebra $KQ$ by the paths of
length $L+1$, where $Q$ is the quiver with a single vertex and a finite number
of loops and $L$ is a positive integer. For any $d>0$, we determine the
irreducible components of the varieties that parametrize the $d$-dimensional
representations of $Lambda$, namely, the components of the classical affine
variety ${
mf{Rep}}_{d}(Lambda)$ and -- equivalently -- those of the
projective parametrizing variety ${
m GRASS}_d(Lambda)$. Our method is to
corner the components by way of a twin pair of upper semicontinuous maps from
${
mf{Rep}}_{d}(Lambda)$ to a poset consisting of sequences of semisimple
modules.
An excerpt of the main result is as follows. Given a sequence ${f S} =
({f S}_0, ..., {f S}_L)$ of semisimple modules with $dim igoplus_{0 le
l le L} {f S}_l = d$, let ${
mf{Rep}}, {f S}$ be the subvariety of
${
mf{Rep}}_{d}(Lambda)$ consisting of the points that parametrize the
modules with radical layering ${f S}$. (The radical layering of a
$Lambda$-module $M$ is the sequence $igl(J^l M / J^{l+1} Migr)_{0 le l
le L}$, where $J$ is the Jacobson radical of $Lambda$.) Suppose the quiver
$Q$ has $r ge 2$ loops. If $d le L+1$, the variety
${
mf{Rep}}_{d}(Lambda)$ is irreducible. If, on the other hand, $d > L+1$,
then the irreducible components of ${
mf{Rep}}_{d}(Lambda)$ are the
closures of the subvarieties ${
mf{Rep}}, {f S}$ for those sequences
${f S}$ which satisfy the inequalities $dim {f S}_l le r dim {f
S}_{l+1}$ and $dim {f S}_{l+1} le r dim {f S}_l$ for $0 le l < L$. As a
byproduct, the main result provides generic information on the modules
corresponding to the irreducible components of the parametrizing varieties. | Source: | arXiv, 1410.3166 | Services: | Forum | Review | PDF | Favorites |
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