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1. An Euler system for GU(2, 1)

   https://doi.org/10.1007/s00208-021-02224-4  

   Loeffler, David; Skinner, Christopher; Zerbes, Sarah Livia (April 2022, Mathematische Annalen)

   Abstract We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of $${{\,\mathrm{GL}\,}}_3$$ GL 3 over imaginary quadratic fields, using the cohomology of Shimura varieties for $${\text {GU}}(2, 1)$$ GU ( 2 , 1 ) . 

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2. Splines on Cayley graphs of the symmetric group

   https://doi.org/10.1017/fms.2025.10037  

   Lesnevich, Nathan_R T (January 2025, Forum of Mathematics, Sigma)

   Abstract A spline is an assignment of polynomials to the vertices of a graph whose edges are labeled by ideals, where the difference of two polynomials labeling adjacent vertices must belong to the corresponding ideal. The set of splines forms a ring. We consider spline rings where the underlying graph is the Cayley graph of a symmetric group generated by a collection of transpositions. These rings generalize the GKM construction for equivariant cohomology rings of flag, regular semisimple Hessenberg and permutohedral varieties. These cohomology rings carry two actions of the symmetric group$$S_n$$whose graded characters are both of general interest in algebraic combinatorics. In this paper, we generalize the graded$$S_n$$-representations from the cohomologies of the above varieties to splines on Cayley graphs of$$S_n$$and then (1) give explicit module and ring generators for whenever the$$S_n$$-generating set is minimal, (2) give a combinatorial characterization of when graded pieces of one$$S_n$$-representation is trivial, and (3) compute the first degree piece of both graded characters for all generating sets. 

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3. The construction of a E7-like quantum subgroup of SU(3)

   https://doi.org/10.1080/00927872.2024.2338216  

   Edie-Michell, Cain; Marinelli, Lance (September 2024, Communications in Algebra)

   In this short note we construct an embedding of the planar algebra for $$\overline{\Rep(U_q(\mathfrak{sl}_3))}$$ at $$q = e^{2\pi i \frac{1}{24}}$$ into the graph planar algebra of di Francesco and Zuber's candidate graph $$\mathcal{E}_4^{12}$$. Via the graph planar algebra embedding theorem we thus construct a rank 11 module category over $$\overline{\Rep(U_q(\mathfrak{sl}_3))}$$ whose graph for action by the vector representation is $$\mathcal{E}_4^{12}$$. This fills a small gap in the literature on the construction of $$\overline{\Rep(U_q(\mathfrak{sl}_3))}$$ module categories. As a consequence of our construction, we obtain the principal graphs of subfactors constructed abstractly by Evans and Pugh. 

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4. Sharp bounds for decomposing graphs into edges and triangles

   https://doi.org/10.1017/S0963548320000358  

   Blumenthal, Adam; Lidický, Bernard; Pehova, Yanitsa; Pfender, Florian; Pikhurko, Oleg; Volec, Jan (March 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract For a real constant α , let $$\pi _3^\alpha (G)$$ be the minimum of twice the number of K 2 ’s plus α times the number of K 3 ’s over all edge decompositions of G into copies of K 2 and K 3 , where K r denotes the complete graph on r vertices. Let $$\pi _3^\alpha (n)$$ be the maximum of $$\pi _3^\alpha (G)$$ over all graphs G with n vertices. The extremal function $$\pi _3^3(n)$$ was first studied by Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $$\pi _3^3(n) \le (1/2 + o(1)){n^2}$$ . We extend their result by determining the exact value of $$\pi _3^\alpha (n)$$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph $${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$$ are the only possible extremal examples for large n . 

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5. Weight 11 Compactly Supported Cohomology of Moduli Spaces of Curves

   https://doi.org/10.1093/imrn/rnad308  

   Payne, Sam; Willwacher, Thomas (January 2024, International Mathematics Research Notices)

   Abstract We study the weight 11 part of the compactly supported cohomology of the moduli space of curves $${\mathcal{M}}_{g,n}$$, using graph complex techniques, with particular attention to the case $n = 0$. As applications, we prove new nonvanishing results for the cohomology of $${\mathcal{M}}_{g}$$, and exponential growth with $$g$$, in a wide range of degrees. 

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