Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symmetric functions in noncommuting variables

Authors: Mercedes H. Rosas and Bruce E. Sagan
Journal: Trans. Amer. Math. Soc. 358 (2006), 215-232
MSC (2000): Primary 05E05; Secondary 05E10, 05A18
Published electronically: December 28, 2004
MathSciNet review: 2171230
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the algebra $\mathbb{Q}\langle \langle x_1,x_2,\ldots\rangle \rangle$ of formal power series in countably many noncommuting variables over the rationals. The subalgebra $\Pi(x_1,x_2,\ldots)$of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties.

References [Enhancements On Off] (What's this?)

  • 1. D. J. Anick, On the homogeneous invariants of a tensor algebra, in ``Algebraic Topology: Proceedings of the International Conference held March 21-24, 1988,'' Mark Mahowald and Stewart Priddy eds., Contemporary Mathematics, Vol. 96, American Math. Society, Providence, RI, 1989, 15-17. MR 90i:55033
  • 2. G. M. Bergman and P. M. Cohn, Symmetric elements in free powers of rings, J. London Math. Soc. (2) 1 (1969), 525-534. MR 40:4301
  • 3. W. Doran and D. Wales, The partition algebra revisited, J. Algebra 231 (2000), 265-330. MR 2001i:16032
  • 4. P. Doubilet, On the foundations of combinatorial theory. VII: Symmetric functions through the theory of distribution and occupancy, Studies in Applied Math. 51 (1972), 377-396. MR 55:2589
  • 5. S. Fomin and C. Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998), 179-200. MR 2000c:05149
  • 6. W. Fulton, ``Young Tableaux,'' London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge, 1999. MR 99f:05119
  • 7. V. Gasharov, Incomparability graphs of $({\mathbf 3}+{\mathbf 1})$-free posets are $s$-positive, Discrete Math. 157 (1996), 193-197. MR 98k:05140
  • 8. D. Gebhard and B. Sagan, A chromatic symmetric function in noncommuting variables, J. Algebraic Combin. 13 (2001), 227-255. MR 2002d:05124
  • 9. I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, J.-I. Thibon, Noncommutative symmetric functions, Adv. in Math. 112 (1995), 218-348.MR 96e:05175
  • 10. I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985), 300-321. MR 87e:05008
  • 11. T. Halverson, Characters of the partition algebra, J. Algebra 238 (2001), 502-533. MR 2002a:20019
  • 12. T. Halverson and J. Fraina, Character orthogonality for the partition algebra and fixed points of permutations. Adv. Appl. Math. 31 (2003), 113-131. MR 2004e:16016
  • 13. V. K. Kharchenko, Algebras of invariants of free algebras, Algebra i Logika 17 (1978) 478-487 (Russian); Algebra and Logic 17 (1978), 316-321 (English translation). MR 80e:16003
  • 14. D. E. Knuth, Permutations, matrices and generalized Young tableaux, Pacific J. Math. 34 (1970), 709-727. MR 42:7535
  • 15. A. Lascoux and M.-P. Schützenberger, Le monoid plaxique, in ``Noncommutative Structures in Algebra and Geometric Combinatorics, (Naples, 1978),'' Quad. Ricerca Sci., Vol. 109, CNR, Rome, 1981, 129-156. MR 83g:20016
  • 16. B. Lindström, On the vector representation of induced matroids, Bull. London Math. Soc. 5 (1973), 85-90. MR 49:95
  • 17. D. E. Littlewood, ``The Theory of Group Characters,'' Oxford University Press, Oxford, 1950. MR 2:3a
  • 18. I. G. Macdonald, ``Symmetric functions and Hall polynomials,'' 2nd edition, Oxford University Press, Oxford, 1995. MR 96h:05207
  • 19. P. A. MacMahon, ``Combinatorial Analysis,'' Vols. 1 and 2, Cambridge University Press, Cambridge, 1915, 1916; reprinted by Chelsea, New York, NY, 1960.
  • 20. P. Martin, The structure of partition algebras, J. Algebra 183 (1996), 319-358. MR 98g:05152
  • 21. S. D. Noble and D. J. A. Welsh, A weighted graph polynomial from chromatic invariants of knots, Symposium à la Mémoire de François Jaeger (Grenoble, 1998) Annales l'Institut Fourier 49 (1999), 1057-1087.MR 2000h:05066
  • 22. G. de B. Robinson, On representations of the symmetric group, Amer. J. Math. 60 (1934), 745-760.
  • 23. M. H. Rosas, MacMahon symmetric functions, the partition lattice, and Young subgroups, J. Combin. Theory Ser. A 96 (2001), 326-340.MR 2002k:05241
  • 24. M. H. Rosas, Specializations of MacMahon symmetric functions and the polynomial algebra, Discrete Math. 246 (2002), 285-293. MR 2003h:05189
  • 25. G.-C. Rota, On the foundations of combinatorial theory I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie 2 (1964), 340-368.MR 30:4688
  • 26. B. Sagan, ``The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions,'' 2nd edition, Springer-Verlag, New York, 2001. MR 2001m:05261
  • 27. C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961), 179-191. MR 22:12047
  • 28. I. Schur, ``Über eine Klasse von Matrizen die sich einer gegebenen Matrix zuordnen lassen,'' Inaugural-Dissertation, Berlin, 1901.
  • 29. R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194. MR 96b:05174
  • 30. R. P. Stanley, ``Enumerative Combinatorics, Volume 1,'' Cambridge University Press, Cambridge, 1997. MR 98a:05001
  • 31. R. P. Stanley, Graph colorings and related symmetric functions: ideas and applications: A description of results, interesting applications, & notable open problems, Selected papers in honor of Adriano Garsia (Taormina, 1994) Discrete Math. 193 (1998), 267-286. MR 2000c:05152
  • 32. R. P. Stanley, ``Enumerative Combinatorics, Volume 2,'' Cambridge University Press, Cambridge, 1999. MR 2000k:05026
  • 33. R. P. Stanley and J. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A 62 (1993), 261-279. MR 94d:05147
  • 34. L. Weisner, Abstract theory of inversion of finite series, Trans. Amer. Math. Soc. 38 (1935), 474-484.
  • 35. H. Whitney, A logical expansion in mathematics, Bull. Amer. Math. Soc. 38 (1932), 572-579.
  • 36. M. C. Wolf, Symmetric functions of noncommuting elements, Duke Math. J. 2 (1936), 626-637.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05E05, 05E10, 05A18

Retrieve articles in all journals with MSC (2000): 05E05, 05E10, 05A18

Additional Information

Mercedes H. Rosas
Affiliation: Departamento de Matemáticas, Universidad Simón Bolívar, Apdo. Postal 89000, Caracas, Venezuela
Address at time of publication: Department of Mathematics & Statistics, York University, Toronto, Ontario, Canada M3J 1P3

Bruce E. Sagan
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Keywords: Noncommuting variables, partition lattice, Schur function, symmetric function
Received by editor(s): October 26, 2002
Received by editor(s) in revised form: January 30, 2004
Published electronically: December 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society