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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Graded Frobenius cluster categories
AU - Grabowski, Jan E.
AU - Pressland, Matthew
PY - 2018/1/1
Y1 - 2018/1/1
N2 - Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other.In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in particular obtaining that degrees are additive on exact sequences.We show that the categories of Buan, Iyama, Reiten and Scott, some of which were used by Geiss, Leclerc and Schroer to categorify cells in partial flag varieties, and those of Jensen, King and Su, categorifying Grassmannians, are examples of graded Frobenius cluster categories.
AB - Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other.In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in particular obtaining that degrees are additive on exact sequences.We show that the categories of Buan, Iyama, Reiten and Scott, some of which were used by Geiss, Leclerc and Schroer to categorify cells in partial flag varieties, and those of Jensen, King and Su, categorifying Grassmannians, are examples of graded Frobenius cluster categories.
KW - math.RT
KW - 13F60 (Primary), 18E30, 16G70 (Secondary)
U2 - 10.25537/dm.2018v23.49-76
DO - 10.25537/dm.2018v23.49-76
M3 - Journal article
VL - 23
SP - 49
EP - 76
JO - Documenta Mathematica
JF - Documenta Mathematica
SN - 1431-0635
ER -