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Defant recently introduced toric promotion, an operator that acts on the labelings of a graph \(G\) and serves as a cyclic analogue of Schützenberger's promotion operator. Toric promotion is defined as the composition of certain toggle operators, listed in a natural cyclic order. We consider more general permutoric promotion operators, which are defined as compositions of the same toggles, but in permuted orders. We settle a conjecture of Defant by determining the orders of all permutoric promotion operators when \(G\) is a path graph. In fact, we completely characterize the orbit structures of these operators, showing that they satisfy the cyclic sieving phenomenon. The first half of our proof requires us to introduce and analyze new broken promotion operators, which can be interpreted via globs of liquid gliding on a path graph. For the latter half of our proof, we reformulate the dynamics of permutoric promotion via stones sliding along a cycle graph and coins colliding with each other on a path graph.Mathematics Subject Classifications: 05E18Keywords: Promotion, toric promotion, Coxeter element, cyclic sieving phenomenon

Let G be a semi-simple simply connected algebraic group over the field ℂ of complex numbers. Let T be a maximal torus of G, and let W be the Weyl group of G with respect to T. Let Z(w, i) be the Bott-Samelson-Demazure-Hansen variety corresponding to a tuple i associated to a reduced expression of an element w ∈ W. We prove that for the tuple i associated to any reduced expression of a minuscule Weyl group element w, the anti-canonical line bundle on Z(w, i) is globally generated. As consequence, we prove that Z(w, i) is weak Fano. Assume that G is a simple algebraic group whose type is different from A2. Let S = {α1, ..., αn} be the set of simple roots. Let w be such that support of w is equal to S. We prove that Z(w, i) is Fano for the tuple i associated to any reduced expression of w if and only if w is a Coxeter element and w − 1 (∑ t = 1 n α t) ∈ − S . [ABSTRACT FROM AUTHOR]

Let N ⊴ G be a pair of finite subgroups of SL2(ℂ) and V a finite-dimensional fundamental G-module. We study Kostant’s generating functions for the decomposition of the SL2(ℂ)-module Sk(V ) restricted to N ◃ G in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined. [ABSTRACT FROM AUTHOR]