## First layer formulas for characters of $\textrm {SL}(n,\textbf {C})$

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- by John R. Stembridge
- Trans. Amer. Math. Soc.
**299**(1987), 319-350 - DOI: https://doi.org/10.1090/S0002-9947-1987-0869415-X
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## Abstract:

Some problems concerning the decomposition of certain characters of $SL(n, {\mathbf {C}})$ are studied from a combinatorial point of view. The specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the Euler characteristic of Hanlon’s so-called “Macdonald complex.” A general recursion is given for computing the irreducible decomposition of these characters. The recursion is explicitly solved for the first layer representations, which are the irreducible representations corresponding to partitions of $n$. In the case of the exterior algebra, this settles a conjecture of Gupta and Hanlon. A further application of the recursion is used to give a family of formal Laurent series identities that generalize the (equal parameter) $q$-Dyson Theorem.## References

- George E. Andrews,
*Problems and prospects for basic hypergeometric functions*, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 191–224. MR**0399528** - D. M. Bressoud and I. P. Goulden,
*Constant term identities extending the $q$-Dyson theorem*, Trans. Amer. Math. Soc.**291**(1985), no. 1, 203–228. MR**797055**, DOI 10.1090/S0002-9947-1985-0797055-8 - Howard Garland and James Lepowsky,
*Lie algebra homology and the Macdonald-Kac formulas*, Invent. Math.**34**(1976), no. 1, 37–76. MR**414645**, DOI 10.1007/BF01418970
G. Z. Giambelli, - P. Hanlon,
*On the decomposition of the tensor algebra of the classical Lie algebras*, Adv. in Math.**56**(1985), no. 3, 238–282. MR**792707**, DOI 10.1016/0001-8708(85)90035-0 - Wim H. Hesselink,
*Characters of the nullcone*, Math. Ann.**252**(1980), no. 3, 179–182. MR**593631**, DOI 10.1007/BF01420081 - Serge Lang,
*Algebra*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0197234**
D. E. Littlewood, - I. G. Macdonald,
*Affine root systems and Dedekind’s $\eta$-function*, Invent. Math.**15**(1972), 91–143. MR**357528**, DOI 10.1007/BF01418931 - I. G. Macdonald,
*Some conjectures for root systems*, SIAM J. Math. Anal.**13**(1982), no. 6, 988–1007. MR**674768**, DOI 10.1137/0513070 - I. G. Macdonald,
*Symmetric functions and Hall polynomials*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR**553598** - R. P. Stanley,
*$\textrm {GL}(n,\textbf {C})$ for combinatorialists*, Surveys in combinatorics (Southampton, 1983) London Math. Soc. Lecture Note Ser., vol. 82, Cambridge Univ. Press, Cambridge, 1983, pp. 187–199. MR**721186** - Richard P. Stanley,
*The stable behavior of some characters of $\textrm {SL}(n,\textbf {C})$*, Linear and Multilinear Algebra**16**(1984), no. 1-4, 3–27. MR**768993**, DOI 10.1080/03081088408817606 - Richard P. Stanley,
*Theory and application of plane partitions. I, II*, Studies in Appl. Math.**50**(1971), 167–188; ibid. 50 (1971), 259–279. MR**325407**, DOI 10.1002/sapm1971503259 - Dennis Stanton,
*Sign variations of the Macdonald identities*, SIAM J. Math. Anal.**17**(1986), no. 6, 1454–1460. MR**860926**, DOI 10.1137/0517103
J. R. Stembridge, - Doron Zeilberger and David M. Bressoud,
*A proof of Andrews’ $q$-Dyson conjecture*, Discrete Math.**54**(1985), no. 2, 201–224. MR**791661**, DOI 10.1016/0012-365X(85)90081-0 - Phil Hanlon,
*Cyclic homology and the Macdonald conjectures*, Invent. Math.**86**(1986), no. 1, 131–159. MR**853448**, DOI 10.1007/BF01391498

*Alcune proprietà delle funzioni simmetriche caratteristiche*, Atti. Torino

**38**(1903), 823-844. R. K. Gupta,

*Polynomial functions on*$g{l_n}({\mathbf {C}})$ (in preparation). R. K. Gupta and P. Hanlon,

*Problem 5 of Problem session*, Combinatorics and Algebra (C. Greene, ed.), Contemporary Math., vol. 34, Amer. Math. Soc., Providence, R. I., 1984.

*The theory of group characters*, 2nd ed., Oxford Univ. Press, 1950.

*Combinatorial decompositions of characters of*$SL(n,\:{\mathbf {C}})$, Ph. D. Thesis, M. I. T., 1985.

## Bibliographic Information

- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**299**(1987), 319-350 - MSC: Primary 20G05; Secondary 05A15, 17B10, 22E46
- DOI: https://doi.org/10.1090/S0002-9947-1987-0869415-X
- MathSciNet review: 869415