JOURNAL BROWSE
Search
Advanced SearchSearch Tips
FREE LIE SUPERALGEBRAS AND THE REPRESENTATIONS OF gl(m, n) AND q(n)
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
FREE LIE SUPERALGEBRAS AND THE REPRESENTATIONS OF gl(m, n) AND q(n)
KWON JAE-HOON;
  PDF(new window)
 Abstract
Let L be the free Lie superalgebra generated by a -graded vector space V over C. Suppose that g is a Lie superalgebra gl(m, n) or q(n). We study the g-module structure on the kth homogeneous component Lk of L when V is the natural representation of g. We give the multiplicities of irreducible representations of g in Lk by using the character of Lk. The multiplicities are given in terms of the character values of irreducible (projective) representations of the symmetric groups.
 Keywords
free lie superalgebra;representation;character;general linear Lie superalgebra;
 Language
English
 Cited by
 References
1.
G. Benkart, Commuting actions-A tale of two groups, Lie algebras and their representations, (Seoul 1995), 1-46, Contemp. Math. 194 Amer. Math. Soc., Providence, RI, 1996

2.
G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, Robert, C. Lee, J. Stroomer, Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994), no. 3, 529-567 crossref(new window)

3.
G. Benkart, C. Lee and A. Ram, Tensor product representations for orthosym- plectic Lie superalgebras, J. Pure Appl. Algebra 130 (1998), no. 1, 1-48 crossref(new window)

4.
A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), no. 2, 118- 175 crossref(new window)

5.
A. Brandt, The free Lie ring and Lie representations of the full linear group, Trans. Amer. Math. Soc. 56 (1944), 528-536 crossref(new window)

6.
R. M. Bryant, Free Lie algebras and formal power series, J. Algebra 253 (2002), no. 1, 167-188 crossref(new window)

7.
R. M. Bryant and R. Stohr, On the module structure of free Lie algebras, Trans. Amer. Math. Soc. 352 (2000), no. 2, 901-934 crossref(new window)

8.
H. Cartan and S. Eilenberg, Homological algebra, Princeton Mathematics Series, Princeton University, 1956

9.
S. Donkin and K. Erdmann, Tilting modules, symmetric functions, and the module structure of the free Lie algeras, J. Algebra 203 (1998), no. 1, 69-90 crossref(new window)

10.
F. G. Frobenius, Uber die Chraktere der symmetrischen Gruppe, Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin (1900), 516-534

11.
F. G. Frobenius, Uber die Chraktere der symmetrischen Gruppe, Gesammelte Abhandlungen, 3 (1968), 148-166

12.
D. B. Fuks, Cohomology of infinite dimensional Lie algebras, Consultant Bureau, New York, 1986

13.
T. Halverson, Characters of the centralizer algebras of mixed tensor represen- tations of GL(r;C) and the quantum group $U_q$(gl(r;C)), Pacific J. Math. 174 (1996), no. 2, 359-410

14.
P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetric groups, Clarendon Press, Oxford, 1992

15.
J. Hong and J. -H. Kwon, Decomposition of free Lie algebras into irreducible components, J. Algebra 197 (1997), no. 1, 127-145 crossref(new window)

16.
T. Jozeflak, Characters of projective representations of symmetric groups, Exposition Math. 7 (1989), 193-247

17.
V. G. Kac, Lie superalgebras, Adv. in Math. 26 (1977), no. 1, 8-96 crossref(new window)

18.
V. G. Kac and S. -J. Kang, Trace formula for graded Lie algebras and monstrous moonshine, Representations of groups, CMS Conf. Proc. 16, Amer. Math. Soc., Providence, RI, 1995, 141-154

19.
S. -J. Kang, Graded Lie superalgebras and the superdimension formula, J. Algebra 204 (1998), no. 2, 597-655 crossref(new window)

20.
S. -J. Kang and J. -H. Kwon, Graded Lie superalgebras, supertrace formula, and orbit Lie superalgebras, Proc. London Math. Soc. 81 (2000), no. 3, 675-724

21.
A. A. Klyachko, Lie elements in the tensor algebra, Siberian Math. J. 15 (1974), no. 6, 914-921 crossref(new window)

22.
W. Kraskiewicz and J. Weyman, Algebra of coinvariants and the action of a Coxeter element, Bayreuth. Math. Schr. 63 (2001), 265-284

23.
J. -H. Kwon, Automorphisms of Borcherds superalgebras and fixed point subalgebras, J. Algebra 259 (2003), no. 2, 533-571 crossref(new window)

24.
D. E. Littlewood, On invariants under restricted groups, Philos. Trans. Roy. Soc. A 239 (1944), 387-417 crossref(new window)

25.
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed. Clarendon Press, Oxford, 1995

26.
A. A. Mikhalev and A. A. Zolotykh, Combinatorial aspects of Lie superalgebras, CRC Press, Boca Raton, FL, 1995

27.
V. M. Petrogradsky, Characters and invariants for free Lie superlagebras, St. Petersburg. Math. J. 13 (2002), no. 1, 107-122

28.
P. Pragacz and A. Thorup, On a Jacobi-Trudi identity for supersymmetric polynomials, Adv. in Math. 95 (1992), no. 1, 8-17 crossref(new window)

29.
A. Ram, Characters of Brauer's centralizer algebras, Pacific J. Math. 169 (1995), no. 1, 173-200 crossref(new window)

30.
C. Reutenauer, Free Lie algebras, Clarendon Press, Oxford, 1993

31.
I. Schur, Uber die Darstellung der symmetrischen und der alternierende Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-250

32.
I. Schur, Uber die rationalen Darstellungen der allgemeinen linearen Gruppe, Preuss. Akad. Wiss. Sitz. 3 (1927), reprinted in Gessamelte Abhandlungen, 68- 85

33.
I. Schur, Uber eine Klasse von Matrizen, die sich einer gegeben Matrix zuordenen lassen, vol. 1, reprinted in Gessamelte Abhandlungen, 1901

34.
A. N. Sergeev, Tensor algebra of the identity representations as a module over the Lie superalgebras Gl(n, m) and Q(n), Math. USSR Sbornik 51 (1985), no. 2, 419-427 crossref(new window)

35.
J. R. Stembridge, A characterization of supersymmetric polynomials, J. Algebra 95 (1985), no. 2, 439-444 crossref(new window)

36.
H. Wenzl, On the structure of Brauer's centralizer algebras, Ann. of Math. 128 (1988), 173-193 crossref(new window)

37.
H. Weyl, Classical groups, Princeton University press, 1946

38.
F. Wever, Uber invarianten von Lie'schen Ringen, Math. Ann. 120 (1949), 563- 580 crossref(new window)

39.
M. Yamaguchi, A duality of the twisted group algebra of the symmetric group and a Lie superalgebra, J. Algebra 222 (1999), no. 1, 301-327 crossref(new window)