Irreducible representations of 𝐴_{𝑛} with a 1-dimensional weight space

Britten, D. J.; Lemire, F. W.

doi:10.1090/S0002-9947-1982-0667158-4

Irreducible representations of $A_{n}$ with a $1$-dimensional weight space
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by D. J. Britten and F. W. Lemire PDF

Trans. Amer. Math. Soc. 273 (1982), 509-540 Request permission

Abstract:

In this paper we classify all irreducible linear representations of the simple Lie algebra ${A_n}$ which admit a one-dimensional weight space with respect to some Cartan subalgebra $H$ of ${A_n}$. We first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the Cartan subalgebra $H$ in the universal enveloping algebra of ${A_n}$ to the base field. We construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of ${A_n}$.

References

Didier Arnal and Georges Pinczon, Sur certaines représentations de l’algèbre de Lie $\textrm {SL}(2)$, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1369–A1372 (French). MR 277577
Richard E. Block, Classification of the irreducible representations of ${\mathfrak {s}}{\mathfrak {l}}(2,\,\textbf {C})$, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 1, 247–250. MR 513751, DOI 10.1090/S0273-0979-1979-14573-7
I. Z. Bouwer, Standard representations of simple Lie algebras, Canadian J. Math. 20 (1968), 344–361. MR 224746, DOI 10.4153/CJM-1968-031-5
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737

Semisimple subalgebras of semisimple Lie algebras

6

Finite dimensional irreducible representations of the unitary and full linear groups and related special functions

64

James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
F. W. Lemire, Irreducible representations of a simple Lie algebra admitting a one-dimensional weight space, Proc. Amer. Math. Soc. 19 (1968), 1161–1164. MR 231872, DOI 10.1090/S0002-9939-1968-0231872-3
F. W. Lemire, Note on weight spaces of irreducible linear representations, Canad. Math. Bull. 11 (1968), 399–403. MR 235073, DOI 10.4153/CMB-1968-045-2
F. W. Lemire, Weight spaces and irreducible representations of simple Lie algebras, Proc. Amer. Math. Soc. 22 (1969), 192–197. MR 243001, DOI 10.1090/S0002-9939-1969-0243001-1
F. W. Lemire, One-dimensional representations of the cycle subalgebra of a semi-simple Lie algebra, Canad. Math. Bull. 13 (1970), 463–467. MR 281765, DOI 10.4153/CMB-1970-085-9

An irreducible representation of

17

F. W. Lemire, A new family of irreducible representations of $A_{n}.$, Canad. Math. Bull. 18 (1975), no. 4, 543–546. MR 422365, DOI 10.4153/CMB-1975-098-4
F. Lemire and M. Pap, $H$-finite irreducible representations of simple Lie algebras, Canadian J. Math. 31 (1979), no. 5, 1084–1106. MR 546961, DOI 10.4153/CJM-1979-100-5
Adolf van den Hombergh, Sur des suites de racines dont la somme des termes est nulle, Bull. Soc. Math. France 102 (1974), 353–364 (French). MR 366998