Reverse plane partitions via representations of quivers Al Garver, - PowerPoint PPT Presentation

Reverse plane partitions via representations of quivers Al Garver, UQAM Ñ University of Michigan (joint with Rebecca Patrias and Hugh Thomas) arXiv: 1812.08345 FPSAC 2019, University of Ljubljana, Slovenia July 4, 2019 1 / 23

Outline minuscule posets Auslander–Reiten quivers nilpotent endomorphisms of quiver representations promotion on reverse plane partitions 2 / 23

A minuscule poset is 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 3 / 23

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 . 4 / 23

Theorem (Proctor ‘84) For any minuscule poset P , the generating function for reverse plane partitions on P is 1 q | ρ | “ ÿ ź 1 ´ q rk p x q ρ : P Ñ Z ě 0 P RPP p P q x P P where | ρ | : “ ř x P P ρ p x q and rk : P Ñ Z ě 1 is the rank function on P . Analogous identities for order filters of certain minuscule posets (Stanley ‘71, Hillman–Grassl ‘76, Gansner ‘81, Pak ‘01, Sulzgruber ‘17) Analogous identities for “skew shapes” (Morales–Pak–Panova ‘15, Naruse–Okada ‘18) 5 / 23

� � � � � � Theorem (Proctor ‘84) For any minuscule poset P , the generating function for reverse plane partitions on P is 1 q | ρ | “ ÿ ź 1 ´ q rk p x q x P P ρ : P Ñ Z ě 0 P RPP p P q where | ρ | : “ ř x P P ρ p x q and rk : P Ñ Z ě 1 is the rank function on P . We will interpret this identity in terms of quiver representations. 4 k » fi – 1 fl 0 3 k » fi – 0 dim p V q “ 1211 k 2 2 fl 1 ” ı 1 1 1 k Q V a quiver a representation of Q dimension vector of V 6 / 23

� � � � � � � � � � � � � � � � � � � � � Any quiver Q has an Auslander–Reiten quiver Γ p Q q whose vertices are the isomorphism classes of indecomposable representations of Q . 4 1111 τ 3 1110 0111 τ τ 2 1100 0110 0011 τ τ τ 1 1000 0100 0010 0001 Γ p Q q - the Auslander–Reiten quiver of Q Q There is a map τ called the Auslander–Reiten translation . The Auslander–Reiten translation partitions the indecomposables into τ -orbits . t vertices of Q u Ð Ñ t τ -orbits u 7 / 23

� � � � � � � � � � � � � � � 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 1111 3 1110 0111 2 1100 0110 0011 1 1000 0100 0010 0001 Γ p Q q - the Auslander–Reiten quiver of Q Q Let C Q , m denote the category of all representations of Q , each of whose indecomposable summands is supported at m . 8 / 23

� � � � � � � � � � V 4 4 V 4 φ 4 � V 3 3 V 3 φ 3 � V 2 2 V 2 φ 2 � V 1 1 V 1 φ 1 Let φ “ p φ i q i P NEnd p V q : “ t nilpotent endomorphisms of V u . Each φ i � λ i “ p λ i r q where partition λ i records the sizes of 1 ě ¨ ¨ ¨ ě λ i the Jordan blocks of φ i . JF p φ q : “ p λ 1 , . . . , λ n q the Jordan form data of φ Theorem (G.–Patrias–Thomas, ‘18) There is a unique maximum value of JF( ¨ ) on NEnd(V) with respect to componentwise dominance order, denoted by GenJF(V). Moreover, it is attained on a dense open subset of NEnd(V). 9 / 23

Theorem (G.–Patrias–Thomas, ‘18) The objects of C Q , m are in bijection with RPP p P Q , m q via V ÞÑ ρ p V q – reverse plane partition from filling the τ -orbits of P Q , m with the Jordan block sizes in GenJF(V) 1111 3 5 4 1110 0 0111 1 5 3 3 ÞÑ 0110 1 0011 1 4 1 2 0010 2 3 1 V ÞÑ ρ p V q Q dim p V q “ 3585 GenJF p V q “ pp 3 q , p 4 , 1 q , p 5 , 3 q , p 5 qq 10 / 23

Theorem (G.–Patrias–Thomas, ‘18) The objects of C Q , m are in bijection with RPP p P Q , m q . 1111 3 5 4 1110 0 0111 1 5 3 3 ÞÑ 0110 1 0011 1 4 1 2 0010 2 3 1 ÞÑ ρ p V q V Q GenJF p V q “ pp 3 q , p 4 , 1 q , p 5 , 3 q , p 5 qq Corollary 1 1 q | ρ | “ q dim p V q “ ÿ ÿ ź ź 1 ´ q dim p V i q “ 1 ´ q rk p x q ρ P RPP p P q V P C Q , m V i P ind p C Q , m q x P P 11 / 23

� � � � � � � � � � � � � � � � � � � Algorithmic construction of ρ p V q Apply the following piecewise linear transformations “from right to left” in Γ p Q q to obtain ρ from V P C Q , m . if W is a summand of V , replace ρ i p V q with ρ i ` 1 p W q “ max W ă U ρ i p U q ` mult p W q , for each V 1 in the τ -orbit of W with W ă V 1 , replace ρ i p V 1 q with ρ i ` 1 p V 1 q “ max V 1 ă U ρ i p U q ` min U ă V 1 ρ i p U q ´ ρ i p V 1 q . 1111 3 0 1110 0 0111 1 0 0 ‚ 0 0 0110 1 0011 1 ‚ ‚ ‚ 0 0010 2 Γ p Q q V 12 / 23

� � � � � � � � � � � � � � � � � � � Algorithmic construction of ρ p V q Apply the following piecewise linear transformations “from right to left” in Γ p Q q to obtain ρ from V P C Q , m . if W is a summand of V , replace ρ i p V q with ρ i ` 1 p W q “ max W ă U ρ i p U q ` mult p W q , for each V 1 in the τ -orbit of W with W ă V 1 , replace ρ i p V 1 q with ρ i ` 1 p V 1 q “ max V 1 ă U ρ i p U q ` min U ă V 1 ρ i p U q ´ ρ i p V 1 q . 1111 3 0 1110 0 0111 1 0 0 0110 1 0011 1 ‚ 0 1 0010 2 ‚ ‚ ‚ 0 V Γ p Q q 13 / 23

� � � � � � � � � � � � � � � � � � � Algorithmic construction of ρ p V q Apply the following piecewise linear transformations “from right to left” in Γ p Q q to obtain ρ from V P C Q , m . if W is a summand of V , replace ρ i p V q with ρ i ` 1 p W q “ max W ă U ρ i p U q ` mult p W q , for each V 1 in the τ -orbit of W with W ă V 1 , replace ρ i p V 1 q with ρ i ` 1 p V 1 q “ max V 1 ă U ρ i p U q ` min U ă V 1 ρ i p U q ´ ρ i p V 1 q . 1111 3 0 1110 0 0111 1 0 0 ‚ 0 1 0110 1 0011 1 ‚ ‚ 3 ‚ 0010 2 V Γ p Q q 14 / 23

� � � � � � � � � � � � � � � � � � � Algorithmic construction of ρ p V q Apply the following piecewise linear transformations “from right to left” in Γ p Q q to obtain ρ from V P C Q , m . if W is a summand of V , replace ρ i p V q with ρ i ` 1 p W q “ max W ă U ρ i p U q ` mult p W q , for each V 1 in the τ -orbit of W with W ă V 1 , replace ρ i p V 1 q with ρ i ` 1 p V 1 q “ max V 1 ă U ρ i p U q ` min U ă V 1 ρ i p U q ´ ρ i p V 1 q . 1111 3 0 1110 0 0111 1 0 2 0110 1 0011 1 ‚ 0 1 0010 2 ‚ ‚ ‚ 3 V Γ p Q q 15 / 23

� � � � � � � � � � � � � � � � � � � Algorithmic construction of ρ p V q Apply the following piecewise linear transformations “from right to left” in Γ p Q q to obtain ρ from V P C Q , m . if W is a summand of V , replace ρ i p V q with ρ i ` 1 p W q “ max W ă U ρ i p U q ` mult p W q , for each V 1 in the τ -orbit of W with W ă V 1 , replace ρ i p V 1 q with ρ i ` 1 p V 1 q “ max V 1 ă U ρ i p U q ` min U ă V 1 ρ i p U q ´ ρ i p V 1 q . 1111 3 0 1110 0 0111 1 0 2 0110 1 0011 1 ‚ 4 1 0010 2 ‚ ‚ ‚ 3 V Γ p Q q 16 / 23

� � � � � � � � � � � � � � � � � � � Algorithmic construction of ρ p V q Apply the following piecewise linear transformations “from right to left” in Γ p Q q to obtain ρ from V P C Q , m . if W is a summand of V , replace ρ i p V q with ρ i ` 1 p W q “ max W ă U ρ i p U q ` mult p W q , for each V 1 in the τ -orbit of W with W ă V 1 , replace ρ i p V 1 q with ρ i ` 1 p V 1 q “ max V 1 ă U ρ i p U q ` min U ă V 1 ρ i p U q ´ ρ i p V 1 q . 1111 3 0 1110 0 0111 1 0 2 ‚ 4 1 0110 1 0011 1 ‚ ‚ ‚ 2 0010 2 Γ p Q q V 17 / 23

� � � � � � � � � � � � � � � � � � � Algorithmic construction of ρ p V q Apply the following piecewise linear transformations “from right to left” in Γ p Q q to obtain ρ from V P C Q , m . if W is a summand of V , replace ρ i p V q with ρ i ` 1 p W q “ max W ă U ρ i p U q ` mult p W q , for each V 1 in the τ -orbit of W with W ă V 1 , replace ρ i p V 1 q with ρ i ` 1 p V 1 q “ max V 1 ă U ρ i p U q ` min U ă V 1 ρ i p U q ´ ρ i p V 1 q . 1111 3 5 1110 0 0111 1 0 2 0110 1 0011 1 ‚ 4 1 ‚ ‚ ‚ 0010 2 2 Γ p Q q V 18 / 23

� � � � � � � � � � � � � � � � � � � Algorithmic construction of ρ p V q Apply the following piecewise linear transformations “from right to left” in Γ p Q q to obtain ρ from V P C Q , m . if W is a summand of V , replace ρ i p V q with ρ i ` 1 p W q “ max W ă U ρ i p U q ` mult p W q , for each V 1 in the τ -orbit of W with W ă V 1 , replace ρ i p V 1 q with ρ i ` 1 p V 1 q “ max V 1 ă U ρ i p U q ` min U ă V 1 ρ i p U q ´ ρ i p V 1 q . 1111 3 5 1110 0 0111 1 5 3 0110 1 0011 1 ‚ 4 1 0010 2 ‚ ‚ ‚ 2 V Γ p Q q 19 / 23