Bijections between invariants associated with indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated with solutions of the Kronecker problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and their corresponding centers involved in the different processes are also given.
Citation: Agustín Moreno Cañadas, Isaías David Marín Gaviria, Pedro Fernando Fernández Espinosa. Brauer configuration algebras and Kronecker modules to categorify integer sequences[J]. Electronic Research Archive, 2022, 30(2): 661-682. doi: 10.3934/era.2022035
Bijections between invariants associated with indecomposable projective modules over some suitable Brauer configuration algebras and invariants associated with solutions of the Kronecker problem are used to categorify integer sequences in the sense of Ringel and Fahr. Dimensions of the Brauer configuration algebras and their corresponding centers involved in the different processes are also given.
| [1] | P. Fahr, C. M. Ringel, A partition formula for Fibonacci numbers, J. Integer Sequences, 11 (2008). Available from: https://emis.dsd.sztaki.hu/journals/JIS/VOL11/Fahr/ringel44.pdf. |
| [2] | P. Fahr, C. M. Ringel, Categorification of the Fibonacci numbers using representations of quivers, preprint, arXiv: 1107.1858. |
| [3] |
P. Fahr, C. M. Ringel, The Fibonacci partition triangles, Adv. Math., 230 (2012), 2513–2535. https://doi.org/10.1016/j.aim.2012.04.010 doi: 10.1016/j.aim.2012.04.010
|
| [4] | C. M. Ringel, The Catalan combinatorics of the hereditary artin algebras, in Recent Developments in Representation Theory, 673 (2016), 51–177. http://dx.doi.org/10.1090/conm/673/13490 |
| [5] |
E. L. Green, S. Schroll, Brauer configuration algebras: A generalization of Brauer graph algebras, Bull. Sci. Math., 121 (2017), 539–572. https://doi.org/10.1016/j.bulsci.2017.06.001 doi: 10.1016/j.bulsci.2017.06.001
|
| [6] | S. Schroll, Brauer graph algebras, inHomological Methods, Representation Theory, and Cluster Algebras, 1 (2018), 177–223. https://doi.org/10.1007/978-3-319-74585-5_6 |
| [7] |
A. Sierra, The dimension of the center of a Brauer configuration algebra, J. Algebra, 510 (2018), 289–318. https://doi.org/10.1016/j.jalgebra.2018.06.002 doi: 10.1016/j.jalgebra.2018.06.002
|
| [8] | D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, $2^{nd}$ edition, Gordon and Breach, 1992. https://zbmath.org/?q=an%3A0818.16009 |
| [9] |
A. G. Zavadskij, On the Kronecker problem and related problems of inear algebra, Linear Algebra Appl., 425 (2007), 26–62. https://doi.org/10.1016/j.laa.2007.03.011 doi: 10.1016/j.laa.2007.03.011
|
| [10] |
A. M. Cañadas, I. D. M. Gaviria, P. F. F. Espinosa, Categorification of some integer sequences via Kronecker modules, JP J. Algebra, Number Theory Appl., 38 (2016), 339–347. https://doi.org/10.17654/NT038040339 doi: 10.17654/NT038040339
|
| [11] | R. Stanley, Enumerative Combinatorics, $2^{nd}$ edition, Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511805967 |
| [12] | G. Andrews, The Theory of Partitions, $2^{nd}$ edition, Cambridge University Press, 1984. https://doi.org/10.1017/CBO9780511608650 |
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Agustín Moreno Cañadas, Isaías David Marín Gaviria, Pedro Fernando Fernández Espinosa. Brauer configuration algebras and Kronecker modules to categorify integer sequences[J]. Electronic Research Archive, 2022, 30(2): 661-682. doi: 10.3934/era.2022035
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