Multiplicities for tensor products on special linear versus classical groups
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Dipendra Prasad
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
· Vinay Wagh
Multiplicities for tensor products on special linear versus classical groups Received: 22 May 2020 / Accepted: 31 October 2020 Abstract. There is a natural bijective correspondence between irreducible (algebraic) selfdual representations of the special linear group with those of classical groups. In this paper, using computations done through the LiE software, we compare tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. More precisely, under the natural correspondence of irreducible finite dimensional selfdual representations of SL2n (C) with those of Spin2n+1 (C), it is easy to see that if the tensor product of three irreducible representations of Spin2n+1 (C) contains the trivial representation, then so does the tensor product of the corresponding representations of SL2n (C). The paper formulates a conjecture in the reverse direction for the pairs (SL2n (C), Spin2n+1 (C)), (SL2n+1 (C), Sp2n (C)), and (Spin2n+2 (C), Sp2n (C)).
Contents 1. 2. 3. 4. 5. 6. 7. 8.
Introduction . . . . . . . . . . . . . . . Relating multiplicities . . . . . . . . . . Review of the group G σ . . . . . . . . . An example of twisted character . . . . . The pair (SL2n (C), Spin2n+1 (C)) . . . . The pair (SL2n+1 (C), Sp2n (C)) . . . . . The case of (Dn+1 , Cn ) . . . . . . . . . Sample computations . . . . . . . . . . Tables representing sample computations References . . . . . . . . . . . . . . . . . .
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1. Introduction There is by now a well-known theory relating irreducible, finite dimensional representations of a group such as SL2n (C) which are selfdual, equivalently, invariant D. Prasad (B): Indian Institute of Technology Bombay, Powai, Mumbai 400076, India. e-mail: [email protected] V. Wagh: Institute of Technology, Guwahati, Guwahati 781039, India. e-mail: [email protected] Mathematics Subject Classification: Primary 22E46; Secondary 20G05
https://doi.org/10.1007/s00229-020-01263-6
D. Prasad, V. Wagh
under the outer automorphism g → θ (g) = tg −1 , with irreducible, finite dimensional representations of Spin2n+1 (C), call this correspondence π SL ↔ π Spin , which has the following character relationship relating character of the representation π of SL2n (C) extended to SL2n (C) θ on the disconnected component with character theory of Spin2n+1 (C): (π SL )(t · θ ) = (π Spin )(t ); here the map t → t is a surjective homomorphism from, say the diagonal torus T in SL2n (C) to t
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