Reverse plane partitions via representations of quivers Al Garver, University of Michigan (joint with Rebecca Patrias and Hugh Thomas) arXiv: 1812.08345 Conference on Geometric Methods in Representation Theory November 24, 2019 1 / 12
Outline nilpotent endomorphisms of quiver representations minuscule posets and Auslander–Reiten quivers reverse plane partitions on minuscule posets periodicity of promotion 2 / 12
� � � � � � � � � Λ “ k Q { I - a finite dimensional algebra, k “ k X “ pp X i q i , p f a q a q P rep p Q , I q » mod Λ φ “ p φ i q i - a nilpotent endomorphism of X NEnd( X ) - all nilpotent endomorphisms of X � X 4 4 X 4 φ 4 � X 3 3 X 3 φ 3 � X 2 2 X 2 φ 2 � X 1 1 X 1 φ 1 Q Lemma The space NEnd(X) is an irreducible algebraic variety. 3 / 12
� � � � � � � � � � X 4 4 X 4 φ 4 � X 3 3 X 3 φ 3 � X 2 2 X 2 φ 2 � X 1 1 X 1 φ 1 For each i , φ i � λ i “ p λ i r q where partition λ i records the sizes 1 ě ¨ ¨ ¨ ě λ i of the Jordan blocks of φ i . JF p φ q : “ p λ 1 , . . . , λ n q the Jordan form data of φ For λ “ p λ 1 ě ¨ ¨ ¨ ě λ r q and λ 1 “ p λ 1 r 1 q , one has λ ď λ 1 in 1 ě ¨ ¨ ¨ ě λ 1 dominance order if λ 1 ` ¨ ¨ ¨ ` λ ℓ ď λ 1 1 ` ¨ ¨ ¨ ` λ 1 ℓ for each ℓ ě 1. Theorem (G.–Patrias–Thomas, ‘18) There is a unique maximum value of JF( ¨ ) on NEnd(X) with respect to componentwise dominance order, denoted by GenJF(X). It is attained on a dense open subset of NEnd(X). 4 / 12
Question For which subcategories C of rep p Q , I q is it the case that any object X P C may be recovered from GenJF(X)? We say such a subcategory is Jordan recoverable . Example Usually GenJF( X ) is not enough information to recover X . Let Q “ 1 Ð 2. 1 X “ k Ð k has GenJF( X ) “ pp 1 q , p 1 qq X 1 “ k 0 Ð k has GenJF( X 1 ) “ pp 1 q , p 1 qq Theorem (G.–Patrias–Thomas ’18) Let Q be a Dynkin quiver and m a minuscule vertex of Q. The category C Q , m of representations of Q all of whose indecomposable summands are supported at m is Jordan recoverable. Moreover, we classify the objects in C Q , m in terms of the combinatorics of the minuscule poset associated with Q and m . 5 / 12
The minuscule posets are defined by choosing a simply-laced Dynkin diagram and a minuscule vertex m . ¨ ¨ ¨ A n 1 2 n n D n 1 2 ¨ ¨ ¨ n ´ 2 n ´ 1 6 E 6 1 2 3 4 5 7 1 2 3 4 5 6 E 7 6 / 12
0 0 1 5 4 1 2 1 3 3 2 5 ρ 3 2 3 4 1 1 2 3 4 4 P A 4 , 3 P D 5 , 1 P D 5 , 4 A reverse plane partition is an order-reversing map ρ : P Ñ Z ě 0 . The objects of C Q , m will be parameterized by reverse plane partitions defined on the minuscule poset associated with Q and m . 7 / 12
� � � � � � � � � � � � � � � � � � Lemma Given a Dynkin quiver Q and a minuscule vertex m, the Hasse quiver of the minscule poset P Q , m is isomorphic to the full subquiver of Γ p Q q on the representations supported at m. 4 1101 0110 0001 3 1110 0101 0010 2 1100 1211 0111 1 1000 0100 1111 Q Γ p Q q - the Auslander–Reiten quiver of Q 8 / 12
� � � � � � � � � � � � � � � � � � � � � � � � � � There is a map τ : Γ p Q q 0 Ñ Γ p Q q 0 called the Auslander–Reiten translation . 4 1101 0110 0001 τ τ 3 1110 0101 0010 τ τ 2 1100 1211 0111 τ τ 1 1000 0100 1111 τ τ Q Γ p Q q The Auslander–Reiten translation partitions the indecomposables into τ -orbits . Q 0 Ð Ñ t τ -orbits u 9 / 12
Theorem (G.–Patrias–Thomas ‘18) The objects of C Q , m are in bijection with RPP p P Q , m q via ÞÑ ρ – reverse plane partition from filling the τ -orbits of X P Q , m with the Jordan block sizes in GenJF(X) 1101 2 2 1110 1 1 ÞÑ 1100 1 1211 0 3 1 1000 0 1111 0 3 1 X ÞÑ ρ p X q 10 / 12
Promotion (pro “ t 4 t 3 t 2 t 1 ) 2 2 2 1 1 1 t 1 t 2 t 4 t 3 ÞÝ Ñ ÞÝ Ñ ÞÝ Ñ 3 1 3 1 8 ´ 1 0 3 1 8 0 8 0 8 ´ 2 8 ´ 2 8 ´ 3 8 ´ 2 8 ´ 3 8 ´ 1 pro pro pro ÞÝ Ñ ÞÝ Ñ ÞÝ Ñ 8 ´ 1 8 ´ 2 8 ´ 3 8 ´ 1 8 ´ 3 0 8 ´ 1 8 ´ 1 8 ´ 3 8 0 0 3 2 2 2 3 1 pro pro ÞÝ Ñ ÞÝ Ñ 8 1 3 2 3 1 8 0 8 1 3 1 Theorem (G.–Patrias–Thomas ‘18) We have pro h “ id where h is the Coxeter number of the root system. 11 / 12
� � � � � � � Thanks! 1110 0 0111 1 0010 2 12 / 12
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