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1. On positivity for the Peterson variety

   Goldin, Rebecca (June 2024, Contemporary Mathematics, American Mathematical Society)

   Tu, Loring W (Ed.)

   We aim in this manuscript to describe a specific notion of geomet- ric positivity that manifests in cohomology rings associated to the flag variety G/B and, in some cases, to subvarieties of G/B. We offer an exposition on the the well-known geometric basis of the homology of G/B provided by Schubert varieties, whose dual basis in cohomology has nonnegative structure constants. In recent work [R. Goldin, L. Mihalcea, and R. Singh, Positivity of Peterson Schubert Calculus, arXiv2106.10372] we showed that the equivariant cohomology of Peterson varieties satisfies a positivity phenomenon similar to that for Schubert calculus for G/B. Here we explain how this positivity extends to this particular nilpotent Hessenberg variety, and offer some open questions about the ingredients for extending positivity results to other Hessenberg varieties. 

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2. Positivity of Peterson Schubert calculus

   https://doi.org/10.1016/j.aim.2024.109879  

   Goldin, Rebecca; Mihalcea, Leonardo; Singh, Rahul (October 2024, Advances in Mathematics)

   Abstract. The Peterson variety is a subvariety of the flag manifold G/B equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert classes indexed by arbitrary Coxeter elements are dual (up to an intersection multiplicity) to the fundamental classes of Peterson cell closures. Dividing these classes by the intersec- tion multiplicities yields a Z-basis for the equivariant cohomology of the Peterson variety. We prove several properties of this basis, including a Graham positivity property for its structure constants, and stability with respect to inclusion in a larger Peterson variety. We also find for- mulae for intersection multiplicities with Peterson classes. This explains geometrically, in arbitrary Lie type, recent positivity statements proved in type A by Goldin and Gorbutt. 

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3. K-theoretic Catalan functions

   https://doi.org/10.1016/j.aim.2022.108421  

   Blasiak, Jonah; Morse, Jennifer; Seelinger, George H (August 2022, Advances in Mathematics)

   We prove that the K-k-Schur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. Lam-Schilling-Shimozono identified the K-k-Schur functions as Schubert representatives for K-homology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the K-k-Schur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for K-k-Schur functions produces a second shift-invariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of Ikeda-Iwao-Maeno, we conjecture that this second basis gives the images of the Lenart-Maeno quantum Grothendieck polynomials under a K-theoretic analog of the Peterson isomorphism. 

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4. Expressing the good in bad times: Examining whether and why positive expressivity in negative contexts affects romantic partners’ responsive support provision.

   https://doi.org/10.1037/pspi0000449  

   Walsh, Rebecca M.; Forest, Amanda L. (January 2024, Journal of Personality and Social Psychology)

   Receiving high-quality, responsive support in times of distress is critical but difficult. In a theoretical review, we previously proposed a process model that explains why support-seekers’ positive expressivity can elicit—but may sometimes suppress—supportive responses from partners (providers) within distress-related contexts. In the current work, we aimed to test direct and indirect pathways linking seeker’s positive expressivity in negative disclosures to provider’s support while addressing notable gaps in the existing literature. Studies considered seeker-expressed positivity as broad, unitary construct (Studies 1, 2, and 4) and explored different types of positivity (Studies 1, 3, and 4): partner-oriented positivity (e.g., gratitude), stressor-oriented positivity (e.g., optimism), and unspecified positivity (e.g., pleasant demeanor). In behavioral observation studies of romantic couples (Studies 1 and 4), seeker-expressed positivity in negative disclosures positively predicted provider responsiveness, even when controlling for seeker-expressed negativity and other plausible third variables. Online experiments with manipulations of seeker-expressed positivity (Studies 2 and 3) yielded causal evidence of positivity’s direct support-eliciting effects. Considering positivity types, partner-oriented positivity and stressor-oriented positivity showed the most robust support-eliciting potential; unspecified positivity also appeared valuable in some contexts. Evidence for several of the model’s indirect pathways emerged in correlational (Study 4) and experimental (Studies 2 and 3) work, providing insights into support-eliciting and support-suppressing mechanisms through which positivity operates. These findings underscore support-seekers’ active role in obtaining support, highlight the value of positive expressivity for eliciting high-quality support, and lay the groundwork for further research on positive expressivity’s effects in support-seeking contexts. 

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5. On two notions of total positivity for partial flag varieties

   Anthony Bloch and Steven Karp (January 2023, Advances in mathematics)

   Lusztig’s total positivity implies Plücker positivity, and it is natural to ask when these two notions of positivity agree. Rietsch (2009) proved that they agree in the case of the Grassmannian Flk;n, and Chevalier (2011) showed that the two notions are distinct for Fl1,3;4. We show that in general, the two notions agree if and only if k1, ..., kl are consecutive integers. 

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