In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system ( W , S ) : a partition of an element w is a subset 𝒫 ⊆ W such that the left inversion set of w is the disjoint union of the left inversion set of the elements in 𝒫 . Partitions of elements of W arises in the study of the Belkale-Kumar product on the cohomology H * ( X , ℤ ) , where X is the complete flag variety of any complex semi-simple algebraic group. Partitions of elements in the symmetric group 𝒮 n are also related to the Babington-Smith model in algebraic statistics or to the simplicial faces of the Littlewood-Richardson cone. Moreover, we state and discuss the conjecture that the number of right descents of w is the sum of the number of right descents of the elements of 𝒫 . In particular: (1) we state an equivalent conjecture in term of the number of atoms in the interval [ e , w ] R of the right weak order, with a word-metric flavour; (2) we give a direct proof that this conjecture holds in the cases of symmetric groups (type A ) and hyperoctahedral groups (type B ).
Some statements in the original publication relied on an implicit assumption. This corrigendum provides the necessary modifications when that assumption does not hold.
Using the connection between translation spreads of the generalized hexagon H ( q ) and linear sets (see Cardinali et al. in Eur J Comb 23:367–376, 2002; Lunardon and Polverino in J Algebraic Comb 18:255–262, 2003), the non-existence of 𝔽 q -translation spreads of H ( q 2 ) when p > 3 , q = p h , and q is large enough is proven. This answers to a question posed in [Marino and Polverino in J Algebraic Comb 42:725-744, 2015].
Jordan schemes generalize association schemes in a similar way as Jordan algebras generalize the associative ones. It is well-known that association schemes of maximal rank are in one-to-one correspondence with groups (so-called thin schemes). In this paper, we classify Jordan schemes of maximal rank-to-order ratio and show that regular Jordan schemes correspond to a special class of Moufang loops, known as Ring Alternative loops.
The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending ideas of Gillespie–Reimer-Berg, we use the RSK correspondence to give a positive interpretation of these counts in terms of the combinatorics of words.
