Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

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
Posted: 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

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. William F. Doran IV and David B. Wales, The partition algebra revisited, J. Algebra 231 (2000), no. 1, 265–330. MR 1779601 (2001i:16032), http://dx.doi.org/10.1006/jabr.2000.8365
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. Sergey Fomin and Curtis Greene, Noncommutative Schur functions and their applications, Discrete Math. 193 (1998), no. 1-3, 179–200. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661368 (2000c:05149), http://dx.doi.org/10.1016/S0012-365X(98)00140-X
6. W. Fulton, ``Young Tableaux,'' London Mathematical Society Student Texts 35, Cambridge University Press, Cambridge, 1999. MR 99f:05119
7. Vesselin Gasharov, Incomparability graphs of (3+1)-free posets are 𝑠-positive, Proceedings of the 6th Conference on Formal Power Series and Algebraic Combinatorics (New Brunswick, NJ, 1994), 1996, pp. 193–197 (English, with English and French summaries). MR 1417294 (98k:05140), http://dx.doi.org/10.1016/S0012-365X(96)83014-7
8. David D. Gebhard and Bruce E. Sagan, A chromatic symmetric function in noncommuting variables, J. Algebraic Combin. 13 (2001), no. 3, 227–255. MR 1836903 (2002d:05124), http://dx.doi.org/10.1023/A:1011258714032
9. Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir S. Retakh, and Jean-Yves Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), no. 2, 218–348. MR 1327096 (96e:05175), http://dx.doi.org/10.1006/aima.1995.1032
10. Ira Gessel and Gérard Viennot, Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985), no. 3, 300–321. MR 815360 (87e:05008), http://dx.doi.org/10.1016/0001-8708(85)90121-5
11. Tom Halverson, Characters of the partition algebras, J. Algebra 238 (2001), no. 2, 502–533. MR 1823772 (2002a:20019), http://dx.doi.org/10.1006/jabr.2000.8613
12. John Farina and Tom Halverson, Character orthogonality for the partition algebra and fixed points of permutations, Adv. in Appl. Math. 31 (2003), no. 1, 113–131. MR 1985823 (2004e:16016), http://dx.doi.org/10.1016/S0196-8858(02)00555-9
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. Paul Martin, The structure of the partition algebras, J. Algebra 183 (1996), no. 2, 319–358. MR 1399030 (98g:05152), http://dx.doi.org/10.1006/jabr.1996.0223
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. Mercedes H. Rosas, MacMahon symmetric functions, the partition lattice, and Young subgroups, J. Combin. Theory Ser. A 96 (2001), no. 2, 326–340. MR 1864127 (2002k:05241), http://dx.doi.org/10.1006/jcta.2001.3186
24. Mercedes H. Rosas, Specializations of MacMahon symmetric functions and the polynomial algebra, Discrete Math. 246 (2002), no. 1-3, 285–293. Formal power series and algebraic combinatorics (Barcelona, 1999). MR 1887491 (2003h:05189), http://dx.doi.org/10.1016/S0012-365X(01)00263-1
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. Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. 111 (1995), no. 1, 166–194. MR 1317387 (96b:05174), http://dx.doi.org/10.1006/aima.1995.1020
30. R. P. Stanley, ``Enumerative Combinatorics, Volume 1,'' Cambridge University Press, Cambridge, 1997. MR 98a:05001
31. Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications: a description of results, interesting applications, & notable open problems, Discrete Math. 193 (1998), no. 1-3, 267–286. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661374 (2000c:05152), http://dx.doi.org/10.1016/S0012-365X(98)00146-0
32. Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282 (2000k:05026)
33. Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, J. Combin. Theory Ser. A 62 (1993), no. 2, 261–279. MR 1207737 (94d:05147), http://dx.doi.org/10.1016/0097-3165(93)90048-D
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.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|