Acyclic Heaps of Pieces, I
R.M. Green
DOI: 10.1023/B:JACO.0000023005.85555.37
Abstract
Heaps of pieces were introduced by Viennot and have applications to algebraic combinatorics, theoretical computer science and statistical physics. In this paper, we show how certain combinatorial properties of heaps studied by Fan and by Stembridge are closely related to the properties of a certain linear map 
E associated to a heap E. We examine the relationship between E and F when F is a subheap of E. This approach allows neat statements and proofs of results on certain associative algebras (generalized Temperley-Lieb algebras) that are otherwise tricky to prove. The key to the proof is to interpret the structure constants of the aforementioned algebras in terms of the maps .
E associated to a heap E. We examine the relationship between E and F when F is a subheap of E. This approach allows neat statements and proofs of results on certain associative algebras (generalized Temperley-Lieb algebras) that are otherwise tricky to prove. The key to the proof is to interpret the structure constants of the aforementioned algebras in terms of the maps .Pages: 173–196
Keywords: heaps of pieces; temperley-Lieb algebras
Full Text: PDF
References
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17. G.X. Viennot, “Heaps of pieces, I: basic definitions and combinatorial lemmas,” Combinatoire Énumérative, G. Labelle and P. Leroux (eds.), Springer-Verlag, Berlin, 1986, pp. 321-350.
2. P. Cartier and D. Foata, “Probl`emes combinatoires de commutation et réarrangements,” Lecture Notes in Mathematics 85 (1969), Springer-Verlag New York/Berlin.
3. T. tom Dieck, “Bridges with pillars: A graphical calculus of knot algebra,” Topology Appl. 78 (1997), 21-38.
4. C.K. Fan, “A Hecke algebra quotient and properties of commutative elements of a Weyl group,” Ph.D. thesis M.I.T. 1995.
5. C.K. Fan, “Structure of a Hecke algebra quotient,” J. Amer. Math. Soc. 10 (1997) 139-167.
6. C.K. Fan and R.M. Green, “On the affine Temperley-Lieb algebras,” Jour. L.M.S. 60 (1999), 366-380.
7. J.J. Graham, “Modular representations of Hecke algebras and related algebras,” Ph.D. thesis University of Sydney, 1995.
8. R.M. Green, “Decorated tangles and canonical bases,” J. Algebra 246 (2001), 594-628.
9. R.M. Green and J. Losonczy, “Canonical bases for Hecke algebra quotients,” Math. Res. Lett. 6 (1999), 213-222.
10. R.M. Green and J. Losonczy, “A projection property for Kazhdan-Lusztig bases,” Int. Math. Res. Not. 1 (2000), 23-34. GREEN
11. D. Kazhdan and G. Lusztig, “Representations of Coxeter groups and Hecke algebras,” Invent. Math. 53 (1979), 165-184.
12. J. Losonczy, “The Kazhdan-Lusztig basis and the Temperley-Lieb quotient in type D,” J. Algebra 233 (2000), 1-15.
13. A. Mazurkiewicz, “Trace theory,” Petri nets, applications and relationship to other models of concurrency Lecture Notes in Computer Science 255, Springer Berlin-Heidelberg-New York, 1987, pp. 279-324.
14. R. Penrose, “Angular momentum: An approach to combinatorial space-time,” Quantum Theory and Beyond E. Bastin, (ed.), Cambridge University Press, Cambridge, 1971, pp. 151-180.
15. J.R. Stembridge, “On the fully commutative elements of Coxeter groups,” J. Algebraic Combin. 5 (1996), 353-385.
16. H.N.V. Temperley and E.H. Lieb, “Relations between percolation and colouring problems and other graph theoretical problems associated with regular planar lattices: Some exact results for the percolation problem,” Proc. Roy. Soc. London Ser. A 322 (1971), 251-280
17. G.X. Viennot, “Heaps of pieces, I: basic definitions and combinatorial lemmas,” Combinatoire Énumérative, G. Labelle and P. Leroux (eds.), Springer-Verlag, Berlin, 1986, pp. 321-350.