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3. Decompositions of Grothendieck Polynomials

   https://doi.org/10.1093/imrn/rnx207  

   Pechenik, Oliver; Searles, Dominic (September 2017, International Mathematics Research Notices)

   null (Ed.)

   Abstract We investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $$K$$-theory of flag varieties (in type $$A$$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $$K$$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $$n$$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $$\beta$$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $$K$$-theory, and we state our results in this more general context. 

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5. Decompositions of Augmented Bergman Complexes

   https://doi.org/10.37236/11192  

   Jeffs, R Amzi (January 2023, The Electronic Journal of Combinatorics)

   The augmented Bergman complex of a closure operator on a finite set interpolates between the order complex of proper flats and the independence complex of the operator. In 2020, Braden, Huh, Matherne, Proudfoot, and Wang showed that augmented Bergman complexes of matroids are always gallery-connected, and recently Bullock, Kelley, Reiner, Ren, Shemy, Shen, Sun, Tao, and Zhang strengthened gallery-connected to shellable by providing two classes of shelling orders: flag-to-basis shellings and basis-to-flag shellings.  We show that augmented Bergman complexes of matroids are vertex decomposable, a stronger property than shellable. We also prove that the augmented Bergman complex of any closure operator is shellable if and only if the order complex of its lattice of flats (that is, its non-augmented Bergman complex) is shellable. As a consequence, an augmented Bergman complex is shellable if and only if it admits a flag-to-basis shelling. Perhaps surprisingly, the same does not hold for basis-to-flag shellings: we describe a closure operator whose augmented Bergman complex is shellable, but has no shelling order with bases appearing first.  

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