On the combinatorics of Kac's asymmetry function

R. M. Green

Commentationes Mathematicae Universitatis Carolinae (2010)

  • Volume: 51, Issue: 2, page 217-235
  • ISSN: 0010-2628

Abstract

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We use categories to recast the combinatorial theory of full heaps, which are certain labelled partially ordered sets that we introduced in previous work. This gives rise to a far simpler set of definitions, which we use to outline a combinatorial construction of the so-called loop algebras associated to affine untwisted Kac--Moody algebras. The finite convex subsets of full heaps are equipped with a statistic called parity, and this naturally gives rise to Kac's asymmetry function. The latter is a key ingredient in understanding the (integer) structure constants of simple Lie algebras with respect to certain Chevalley bases, which also arise naturally in the context of heaps.

How to cite

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Green, R. M.. "On the combinatorics of Kac's asymmetry function." Commentationes Mathematicae Universitatis Carolinae 51.2 (2010): 217-235. <http://eudml.org/doc/37754>.

@article{Green2010,
abstract = {We use categories to recast the combinatorial theory of full heaps, which are certain labelled partially ordered sets that we introduced in previous work. This gives rise to a far simpler set of definitions, which we use to outline a combinatorial construction of the so-called loop algebras associated to affine untwisted Kac--Moody algebras. The finite convex subsets of full heaps are equipped with a statistic called parity, and this naturally gives rise to Kac's asymmetry function. The latter is a key ingredient in understanding the (integer) structure constants of simple Lie algebras with respect to certain Chevalley bases, which also arise naturally in the context of heaps.},
author = {Green, R. M.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lie algebra; Chevalley basis; heap; Lie algebra; Chevalley basis; heap},
language = {eng},
number = {2},
pages = {217-235},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the combinatorics of Kac's asymmetry function},
url = {http://eudml.org/doc/37754},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Green, R. M.
TI - On the combinatorics of Kac's asymmetry function
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 2
SP - 217
EP - 235
AB - We use categories to recast the combinatorial theory of full heaps, which are certain labelled partially ordered sets that we introduced in previous work. This gives rise to a far simpler set of definitions, which we use to outline a combinatorial construction of the so-called loop algebras associated to affine untwisted Kac--Moody algebras. The finite convex subsets of full heaps are equipped with a statistic called parity, and this naturally gives rise to Kac's asymmetry function. The latter is a key ingredient in understanding the (integer) structure constants of simple Lie algebras with respect to certain Chevalley bases, which also arise naturally in the context of heaps.
LA - eng
KW - Lie algebra; Chevalley basis; heap; Lie algebra; Chevalley basis; heap
UR - http://eudml.org/doc/37754
ER -

References

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  9. McGregor-Dorsey Z.S., Full heaps over Dynkin diagrams of type A ˜ , M.A. thesis, University of Colorado at Boulder, 2008. 
  10. Stembridge J.R., 10.1006/jabr.2000.8488, J. Algebra 235 (2001), 722–743. Zbl0973.17034MR1805477DOI10.1006/jabr.2000.8488
  11. Vavilov N.A., 10.1090/S1061-0022-08-01008-X, St Petersburg Math. J. 19 (2008), 519–543. MR2381932DOI10.1090/S1061-0022-08-01008-X
  12. Viennot G.X., Heaps of pieces, I: basic definitions and combinatorial lemmas, in Combinatoire Énumérative (ed. G. Labelle and P. Leroux), Springer, Berlin, 1986, pp. 321–350. Zbl0792.05012MR0927773
  13. Wildberger N.J., 10.1016/S0196-8858(02)00541-9, Adv. Appl. Math. 30 (2003), 385–396. Zbl1023.17015MR1979800DOI10.1016/S0196-8858(02)00541-9

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