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1. 0-Hecke modules for row-strict dual immaculate functions

   https://doi.org/10.1090/tran/9006  

   Niese, Elizabeth; Sundaram, Sheila; van_Willigenburg, Stephanie; Vega, Julianne; Wang, Shiyun (February 2024, Transactions of the American Mathematical Society)

   We introduce a new basis of quasisymmetric functions, the row-strict dual immaculate functions. We construct a cyclic, indecomposable 0-Hecke algebra module for these functions. Our row-strict immaculate functions are related to the dual immaculate functions of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15) by the involution $\mathrm{\psi <\#comment/>}$on the ring $\mathrm{QSym}$of quasisymmetric functions. We give an explicit description of the effect of $\mathrm{\psi <\#comment/>}$on the associated 0-Hecke modules, via the poset induced by the 0-Hecke action on standard immaculate tableaux. This remarkable poset reveals other 0-Hecke submodules and quotient modules, often cyclic and indecomposable, notably for a row-strict analogue of the extended Schur functions studied in Assaf-Searles (2019). Like the dual immaculate function, the row-strict dual immaculate function is the generating function of a suitable set of tableaux, corresponding to a specific descent set. We give a complete combinatorial and representation-theoretic picture by constructing 0-Hecke modules for the remaining variations on descent sets, and showing thatallthe possible variations for generating functions of tableaux occur as characteristics of the 0-Hecke modules determined by these descent sets. 

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2. Cooperation, domination: Twin functions of third‐party punishment

   https://doi.org/10.1111/spc3.12992  

   Wylie, Jordan; Gantman, Ana (July 2024, Social and Personality Psychology Compass)

   Abstract Rules serve many important functions in society. One such function is to codify, and make public and enforceable, a society's desired prescriptions and proscriptions. This codification means that rules come with predefined punishments administered by third parties. We argue that when we look at how third parties punish rule violations, we see that rules and their punishments often serve dual functions. They support and help to maintain cooperation as it is usually theorized, but they also facilitate the domination of marginalized others. We begin by reviewing literature on rules and third‐party punishment, arguing that a great deal of punishment research has neglected to consider the unique power of codified rules. We also argue that by focusing on codified rules, it becomes clear that the enforcement of such rules via third‐party punishment is often used to exert control, punish retributively, and oppress outgroup members. By challenging idealized theory of rules as facilitators of social harmony, we highlight their role in satisfying personal punishment motives, and facilitating discrimination in a way that is uniquely justifiable to those who enforce them. 

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3. Inhomogeneous spin q -Whittaker polynomials

   https://doi.org/10.5802/afst.1761  

   Borodin, Alexei; Korotkikh, Sergei (July 2024, Annales de la Faculté des sciences de Toulouse : Mathématiques)

   We introduce and study an inhomogeneous generalization of the spin $q$-Whittaker polynomials from [15]. These are symmetric polynomials, and we prove a branching rule, skew dual and non-dual Cauchy identities, and an integral representation for them. Our main tool is a novel family of deformed Yang–Baxter equations. 

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4. Special cubulation of strict hyperbolization

   https://doi.org/10.1007/s00222-024-01241-9  

   Lafont, Jean-François; Ruffoni, Lorenzo (June 2024, Inventiones mathematicae)

   Abstract We prove that the Gromov hyperbolic groups obtained by the strict hyperbolization procedure of Charney and Davis are virtually compact special, hence linear and residually finite. Our strategy consists in constructing an action of a hyperbolized group on a certain dual$$\operatorname {CAT}(0)$$ $CAT\left(0\right)$cubical complex. As a result, all the common applications of strict hyperbolization are shown to provide manifolds with virtually compact special fundamental group. In particular, we obtain examples of closed negatively curved Riemannian manifolds whose fundamental groups are linear and virtually algebraically fiber. 

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5. RANK AND BORDER RANK OF KRONECKER POWERS OF TENSORS AND STRASSEN’S LASER METHOD

   https://doi.org/10.1007/s00037-021-00217-y  

   Conner, Austin; Gesmundo, Fulvio; Landsberg, Joseph M.; Ventura, Emanuele (April 2022, Computational complexity)

   We prove that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor Tcw,q is the square of its border rank for q > 2 and that the border rank of its Kronecker cube is the cube of its border rank for q > 4. This answers questions raised implicitly by Coppersmith & Winograd (1990, §11) and explicitly by Bl¨aser (2013, Problem 9.8) and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith–Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen’s laser method, introducing a skew-symmetric version of the Coppersmith– Winograd tensor, Tskewcw,q. For q = 2, the Kronecker square of this tensor coincides with the 3 × 3 determinant polynomial, det3 ∈ C9 ⊗ C9 ⊗ C9, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of det3, exhibiting a strict submultiplicative behaviour for Tskewcw,2 which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in C3 ⊗ C3 ⊗ C3. 

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