Formulas for multi-parameter classes of Kazhdan–Lusztig polynomials in S(n)

We give explicit formulas for several classes of Kazhdan–Lusztig polynomials of the symmetric group which are related to others already considered in the literature. In particular, we generalize two theorems of Brenti and Simion [Explicit formulae for some Kazhdan–Lusztig polynomials, J. Algebraic Combin. 11 (2000) 187–196], we prove an unpublished conjecture of Brenti, and we treat the case missing in Theorem 4.8 of F. Caselli [Proof of two conjectures of Brenti and Simion on Kazhdan–Lusztig polynomials, J. Algebraic Combin. 18 (2003) 171–187].     

On difference sets with small $$\lambda $$ λ

Journal of Algebraic Combinatorics - In a 1989 paper, Arasu (Arch Math 53:622–624, 1989) used an observation about multipliers to show that no (352, 27, 2) difference set...     

Complete bipartite multi-graphs with a unique regular dessin

Journal of Algebraic Combinatorics - A regular dessin is an orientable edge-regular bipartite map. Jones et al.(J Combin Theory Ser B 98:241–248, 2008) showed that a complete bipartite graph...     

The enumeration of extreme rigid honeycombs

Journal of Algebraic Combinatorics - Rigid honeycombs were introduced by Knutson et al. (J Am Math Soc 17:19–48, 2004), and they were shown in Bercovici et al. (J Funct Anal...     

Algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor

Algebraic Ricci solitons on Lie groups with left-invariant (pseudo)Riemannian metric and zero Schouten–Weyl tensor are studied. The absence of nontrivial algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor and diagonalizable Ricci operator is proved.     

Skew Pieri rules for Hall–Littlewood functions

We produce skew Pieri rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (J. Comb. Theory Ser. A 118(1):277–290, 2011). The first two were conjectured by the first author (Konvalinka in J. Algebraic Comb. 35(4):519–545, 2012). The key ingredients in the proofs are a q-binomial identity for skew partitions and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and the antipode.     

On balanced Cayley maps over abelian groups

A notion of a balanced automorphism of a group was introduced in [J. Sirǎn and M. Skoviera. Regular maps from Cayley graphs. I. Balanced Cayley maps. Algebraic graph theory (Leibnitz, 1989). Discrete Math. 109 (1992), no. 1–3, 265–276] in order to study regular Cayley maps over finite groups. One of the open questions formulated in the paper was this: which finite abelian groups admit a balanced automorphism? The presented paper answers this question.     

An iterative formula for the Kostka–Foulkes polynomials

Journal of Algebraic Combinatorics - An iterative formula for the Kostka–Foulkes polynomials is given using the vertex operator realization of the Hall–Littlewood polynomials. The...     

Non-hyperoctahedral categories of two-colored partitions part I: new categories

Journal of Algebraic Combinatorics - Compact quantum groups can be studied by investigating their representation categories in analogy to the Schur–Weyl/Tannaka–Krein approach. For the...     

Algebraic theory of difference equations and Mahler functions

Algebraic independence of certain Mahler functions constructed from Rudin–Schapiro sequences and Baum–Sweet sequences is proved, using difference Riccati equations and the notion of difference field extension of valuation ring type.     

A seminormal form for partition algebras

Using a new presentation for partition algebras [J. Algebraic Combin. 37 (3) (2013) 401–454], we derive explicit combinatorial formulae for the seminormal representations of the partition algebras. These results generalise to the partition algebras the classical formulae given by Young for the symmetric group.     

Fiber cones of rational normal scrolls are Cohen–Macaulay

Journal of Algebraic Combinatorics - In this paper, we show that the fiber cones of rational normal scrolls are Cohen–Macaulay. As an application, we compute their Castelnuovo–Mumford...     

Nowhere-zero 3-flows in Cayley graphs on generalized dihedral group and generalized quaternion group

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111: 416–419]. We also generalizes an early result of M. Nánásiová and M. ?koviera [J. Algebraic Combin., 2009, 30: 103–110].     

Quantitative Morse–Sard Theorem via Algebraic Lemma

We give a short proof of the so-called Quantitative Morse–Sard Theorem as an application of Gromov?s Algebraic Lemma.     

Réalisation ℓ-adique des motifs triangulés géométriques II

For geometrical triangulated motives with rational coefficients over a ground field of characteristic zero which is embeddable into \mathbbC{\mathbb{C}} , Huber (J. Algebraic Geom. 9(4):755–799, 2000; J. Algebraic Geom. 13(1):195–207, 2004) has constructed a realization functor with values in the category of mixed realization of Huber (Mixed motives and their realization in derived categories. Lecture Notes in Mathematics, vol. 1604. Springer, Berlin, 1995). In this sequel to Ivorra (Doc Math 12:607–671, 2007), we prove that the ℓ-adic realization functor obtained in Theorem 4.3 of Ivorra (Doc Math 12:607–671, 2007) is the same up to a canonical isomorphism as the ℓ-adic component of A. Huber’s construction. In this way (Ivorra in Doc Math 12:607–671, 2007) might be viewed as an integral generalization to all noetherian separed schemes of the work (Huber in J. Algebraic Geom. 9(4):755–799, 2000; J. Algebraic Geom. 13(1):195–207, 2004) as far as the ℓ-adic setting is concerned. We also prove a comparison theorem with the classical ℓ-adic cycle class map over a perfect field using a naive motivic cycle class map.     

Maximal generating degrees of integral closures of powers of monomial ideals

Journal of Algebraic Combinatorics - We give effective lower and upper bounds on the maximal generating degree $$d(\overline{I^n})$$ and the Castelnuovo–Mumford regularity...     

Weighted infinitesimal unitary bialgebras of rooted forests,symmetric cocycles and pre-Lie algebras

Journal of Algebraic Combinatorics - The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we...     

Lattice of dominant weights of affine Kac–Moody algebras

Journal of Algebraic Combinatorics - The dual space of the Cartan subalgebra in a Kac–Moody algebra has a partial ordering defined by the rule that two elements are related if and only if...     

A note on the short algebraic proof of Farkas' Lemma

The author published ‘A Short Algebraic Proof of the Farkas Lemma’ [SIAM J. Optim. 19 (2008), pp. 234–239]. The author then found, in his opinion, a better exposition of the proof. He would therefore like to publish the new form of the proof in this note.     

The Terwilliger algebra of the halved folded 2n-cube from the viewpoint of its automorphism group action

Journal of Algebraic Combinatorics - Let $$\frac{1}{2}{\overline{H}}(2n,2)$$ denote the halved folded 2n-cube with vertex set X and let $$T{:}{=}T(x)$$ denote the Terwilliger algebra of...