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1. The q-analog of Kostant’s partition function and the highest root of the simple Lie algebras

   Harris, Pamela; Insko, Erik; Omar, Mo (April 2018, The Australasian journal of combinatorics)

   For a given weight of a complex simple Lie algebra, the q-analog of Kostant’s partition function is a polynomial valued function in the variable q, where the coefficient of qk is the number of ways the weight can be written as a nonnegative integral sum of exactly k positive roots. In this paper we determine generating functions for the q-analog of Kostant’s partition function when the weight in question is the highest root of the classical Lie algebras of types B, C, and D, and the exceptional Lie algebras of type G2, F4, E6, E7, and E8. 

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2. Super major index and Thrall’s problem

   https://doi.org/10.5802/alco.421  

   Armon, Sam; Swanson, Joshua P (June 2025, Algebraic Combinatorics)

   Thrall’s problem asks for the Schur decomposition of the higher Lie modules $$L_\lambda$$, which are defined using the free Lie algebra and decompose the tensor algebra as a general linear group module. Although special cases have been solved, Thrall’s problem remains open in general. We generalize Thrall’s problem to the free Lie superalgebra, and prove extensions of three known results in this setting: Brandt’s formula, Klyachko’s identification of the Schur–Weyl dual of $$L_n$$, and Kráskiewicz–Weyman’s formula for the Schur decomposition of $$L_n$$. The latter involves a new version of the major index on super tableaux, which we show corresponds to a $q,t$-hook formula of Macdonald. 

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3. A Bayesian Analysis of Physical Parameters for 783 Kepler Close Binaries: Extreme-mass-ratio Systems and a New Mass Ratio versus Period Lower Limit

   https://doi.org/10.3847/1538-4365/ac75bd  

   Kobulnicky, Henry A.; Molnar, Lawrence A.; Cook, Evan M.; Henderson, Lauren E. (August 2022, The Astrophysical Journal Supplement Series)

   Abstract Contact binary star systems represent the long-lived penultimate phase of binary evolution. Population statistics of their physical parameters inform an understanding of binary evolutionary pathways and end products. We use light curves and new optical spectroscopy to conduct a pilot study of ten (near) contact systems in the long-period (P> 0.5 days) tail of close binaries in the Kepler field. We use PHOEBE light-curve models to compute Bayesian probabilities on five principal system parameters. Mass ratios and third-light contributions measured from spectra agree well with those inferred from the light curves. Pilot study systems have extreme mass ratiosq< 0.32. Most are triples. Analysis of the unbiased sample of 783 0.15 d <P< 2 days (near) contact binaries results in 178 probable contact systems, 114 probable detached systems, and 491 ambiguous systems for which we report best-fitting and 16th-/50th-/84th-percentile parameters. Contact systems are rare at periodsP> 0.5 days, as are systems withq> 0.8. There exists an empirical mass ratio lower limit $q_{\min }\left(P\right)$≈ 0.05–0.15 below which contact systems are absent, supporting a new set of theoretical predictions obtained by modeling the evolution of contact systems under the constraints of mass and angular momentum conservation. Premerger systems should lie at long periods and near this mass ratio lower limit, which rises fromq= 0.044 forP= 0.74 days toq= 0.15 atP= 2.0 days. These findings support a scenario whereby nuclear evolution of the primary (more massive) star drives mass transfer to the primary, thus moving systems toward extremeqand largerPuntil the onset of the Darwin instability at $q_{\min }$precipitates a merger. 

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4. Prescribing $$Q$$-curvature on even-dimensional manifolds with conical singularities

   https://doi.org/10.4171/RMI/1543  

   Jevnikar, Aleks; Sire, Yannick; Yang, Wen (February 2025, Revista Matemática Iberoamericana)

   On a2m-dimensional closed manifold, we investigate the existence of prescribedQ-curvature metrics with conical singularities. We present here a general existence and multiplicity result in the supercritical regime. To this end, we first carry out a blow-up analysis of a2mth-order PDE associated to the problem, and then apply a variational argument of min-max type. Form>1, this seems to be the first existence result for supercritical conic manifolds different from the sphere. 

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5. The weight hierarchy of decreasing norm-trace codes

   https://doi.org/10.1007/s10623-025-01619-7  

   Camps-Moreno, Eduardo; López, Hiram H; Matthews, Gretchen L; San-José, Rodrigo (April 2025, Designs, Codes and Cryptography)

   Abstract The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II,t-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating sets of monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by$$x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$$ $x^u=y^{q^{s-1}}+y^{q^{s-2}}+\cdots +y$over the finite field of cardinality$$q^s$$ $q^s$, whereuis a positive divisor of$$\frac{q^s - 1}{q - 1}$$ $\frac{q^s-1}{q-1}$. As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters. 

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