Deligne categories and reduced Kronecker coefficients
The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions λ,μ,τ of n, the multiplicity of λ in μ⊗τ is called the Kronecker coefficient g[superscript λ][subscript μ,τ]. When the first part of each of th...
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| Format: | Article |
| Language: | English |
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Springer US
2016
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| Online Access: | http://hdl.handle.net/1721.1/105783 https://orcid.org/0000-0002-0226-1859 |
| Summary: | The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions λ,μ,τ of n, the multiplicity of λ in μ⊗τ is called the Kronecker coefficient g[superscript λ][subscript μ,τ]. When the first part of each of the partitions is taken to be very large (the remaining parts being fixed), the values of the appropriate Kronecker coefficients stabilize; the stable value is called the reduced (or stable) Kronecker coefficient. These coefficients also generalize the Littlewood–Richardson coefficients and have been studied quite extensively. In this paper, we show that reduced Kronecker coefficients appear naturally as structure constants of Deligne categories [bar under Rep](S[subscript t]). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories [bar under Rep](S[subscript t]) and derive new combinatorial identities. |
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