Classifying representations by way of Grassmannians

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with fixed dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and fixed squarefree top <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In the case of existence of a moduli space—unexpectedly frequent in light of the stringency of fine classification—this space is always projective and, in fact, arises as a closed subvariety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper G German r German a German s German s Subscript d Superscript upper T"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">G</mml:mi> <mml:mi mathvariant="fraktur">r</mml:mi> <mml:mi mathvariant="fraktur">a</mml:mi> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">s</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>T</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\operatorname {\mathfrak {Grass}}^T_d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper G German r German a German s German s Subscript d Superscript upper T"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">G</mml:mi> <mml:mi mathvariant="fraktur">r</mml:mi> <mml:mi mathvariant="fraktur">a</mml:mi> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">s</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>T</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\operatorname {\mathfrak {Grass}}^T_d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of ‘finite local representation type at a given simple <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ’, the radical layering <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper J Superscript l Baseline upper M slash upper J Superscript l plus 1 Baseline upper M right-parenthesis Subscript l greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-OPEN"> <mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>l</mml:mi> </mml:mrow> </mml:msup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>J</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>l</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>M</mml:mi> <mml:msub> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-CLOSE"> <mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>l</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\bigl ( J^{l}M/ J^{l+1}M \bigr )_{l \ge 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is shown to be a classifying invariant for the modules with top <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This re