Multiplicities for tensor products on special linear versus classical groups

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 \(\mathrm{SL}_{2n}({\mathbb {C}})\) with those of \(\mathrm{Spin}_{2n+1}({\mathbb {C}})\), it is easy to see that if the tensor product of three irreducible representations of \(\mathrm{Spin}_{2n+1}({\mathbb {C}})\) contains the trivial representation, then so does the tensor product of the corresponding representations of \(\mathrm{SL}_{2n}({\mathbb {C}})\). The paper formulates a conjecture in the reverse direction for the pairs \((\mathrm{SL}_{2n}({\mathbb {C}}), \mathrm{Spin}_{2n+1}({\mathbb {C}})), (\mathrm{SL}_{2n+1}({\mathbb {C}}), \mathrm{Sp}_{2n}({\mathbb {C}})),\) and \( (\mathrm{Spin}_{2n+2}({\mathbb {C}}), \mathrm{Sp}_{2n}({\mathbb {C}})) \).

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Acknowledgements

The authors thank CS Rajan for bringing the paper of Kapovitch [8] and that of Kumar [11] to their notice. The authors also thank J. Hong, S. Kumar and N. Ressayre for very useful comments on this work. The second author thanks Kalpesh Kapoor for his help in computational aspects. The work of the first author was supported by the JC Bose National Fellowship of the Govt. of India, project number JBR/2020/000006, and also supported by a grant of the Government of the Russian Federation for the state support of scientific research carried out under the agreement 14.W03.31.0030 dated 15.02.2018.

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Correspondence to Dipendra Prasad.

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Prasad, D., Wagh, V. Multiplicities for tensor products on special linear versus classical groups. manuscripta math. (2020). https://doi.org/10.1007/s00229-020-01263-6

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Mathematics Subject Classification

  • Primary 22E46
  • Secondary 20G05