The main purpose of this thesis is to obtain surprising identities by counting the representations of quivers over finite fields. A classical result states that the dimension vectors of the absolutely indecomposable representations of a quiver ?? are in one-to-one correspondence with the positive roots of a root system ??, which is infinite in general. For a given dimension vector ?? ??? ??+, the number A??(??, q), which counts the isomorphism classes of the absolutely indecomposable representations of ?? of dimension ?? over the finite field Fq, turns out to be a polynomial in q with integer coefficients, which have been conjectured to be nonnegative by Kac. The main result of this thesis is a multi-variable formal identity which expresses an infinite series as a formal product indexed by ??+ which has the coefficients of various polynomials A??(??, q) as exponents. This identity turns out to be a qanalogue of the remarkable Weyl-Macdonald-Kac denominator identity modulus a conjecture of Kac, which asserts that the multiplicity of ?? is equal to the constant term of A??(??, q). An equivalent form of this conjecture is established and a partial solution is obtained. A new proof of the integrality of A??(??, q) is given. Three Maple programs have been included which enable one to calculate the polynomials A??(??, q) for quivers with at most three nodes. All sample out-prints are consistence with Kac???s conjectures. Another result of this thesis is as follows. Let A be a finite dimensional algebra over a perfect field K, M be a finitely generated indecomposable module over A ???K ??K. Then there exists a unique indecomposable module M??? over A such that M is a direct summand of M??? ???K ??K, and there exists a positive integer s such that Ms = M ??? ?? ?? ?? ??? M (s copies) has a unique minimal field of definition which is isomorphic to the centre of End ??(M???) rad (End ??(M???)). If K is a finite field, then s can be taken to be 1.
| Identifer | oai:union.ndltd.org:ADTP/243027 |
| Date | January 1998 |
| Creators | Hua, Jiuzhao , Mathematics & Statistics, Faculty of Science, UNSW |
| Publisher | Awarded by:University of New South Wales. Mathematics & Statistics |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | Copyright Hua Jiuzhao ., http://unsworks.unsw.edu.au/copyright |
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