We show that in a finite tensor category, the tensor product property holds for support varieties if and only if it holds between indecomposable periodic objects. We apply this result to deduce the tensor product property for a large class of categories, those of modules for skew group algebras formed by exterior algebras with certain finite group actions. These include the symmetric finite tensor categories over algebraically closed fields of characteristic zero, thus giving a new proof of the tensor product property for these categories.
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Keywords: finite tensor categories, support varieties, tensor product property
Bergh, Petter Andreas 1; Plavnik, Julia Yael 2; Witherspoon, Sarah 3
@article{ART_2024__1_4_539_0, author = {Bergh, Petter Andreas and Plavnik, Julia Yael and Witherspoon, Sarah}, title = {Support varieties for finite tensor categories: the tensor product property}, journal = {Annals of Representation Theory}, pages = {539--566}, publisher = {The Publishers of ART}, volume = {1}, number = {4}, year = {2024}, doi = {10.5802/art.18}, language = {en}, url = {https://art.centre-mersenne.org/articles/10.5802/art.18/} }
TY - JOUR AU - Bergh, Petter Andreas AU - Plavnik, Julia Yael AU - Witherspoon, Sarah TI - Support varieties for finite tensor categories: the tensor product property JO - Annals of Representation Theory PY - 2024 SP - 539 EP - 566 VL - 1 IS - 4 PB - The Publishers of ART UR - https://art.centre-mersenne.org/articles/10.5802/art.18/ DO - 10.5802/art.18 LA - en ID - ART_2024__1_4_539_0 ER -
%0 Journal Article %A Bergh, Petter Andreas %A Plavnik, Julia Yael %A Witherspoon, Sarah %T Support varieties for finite tensor categories: the tensor product property %J Annals of Representation Theory %D 2024 %P 539-566 %V 1 %N 4 %I The Publishers of ART %U https://art.centre-mersenne.org/articles/10.5802/art.18/ %R 10.5802/art.18 %G en %F ART_2024__1_4_539_0
Bergh, Petter Andreas; Plavnik, Julia Yael; Witherspoon, Sarah. Support varieties for finite tensor categories: the tensor product property. Annals of Representation Theory, Volume 1 (2024) no. 4, pp. 539-566. doi : 10.5802/art.18. https://art.centre-mersenne.org/articles/10.5802/art.18/
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