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  • 1.
    Brubaker, Ben
    et al.
    School of Mathematics, University of Minnesota, MN, Minneapolis, United States.
    Buciumas, Valentin
    Department of Mathematics, Pohang University of Science and Technology, Pohang, South Korea.
    Bump, Daniel
    Department of Mathematics, Stanford University, CA, Stanford, United States.
    Gustafsson, Henrik P. A.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Iwahori-metaplectic duality2024In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 109, no 6, article id e12896Article in journal (Refereed)
    Abstract [en]

    We construct a family of solvable lattice models whose partition functions include p-adic Whittaker functions for general linear groups from two very different sources: from Iwahori-fixed vectors and from metaplectic covers. Interpolating between them by Drinfeld twisting, we uncover unexpected relationships between Iwahori and metaplectic Whittaker functions. This leads to new Demazure operator recurrence relations for spherical metaplectic Whittaker functions. In prior work of the authors it was shown that the row transfer matrices of certain lattice models for spherical metaplectic Whittaker functions could be represented as ‘half-vertex operators’ operating on the q-Fock space of Kashiwara, Miwa and Stern. In this paper the same is shown for all the members of this more general family of lattice models including the one representing Iwahori Whittaker functions.

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  • 2.
    Falgas-Ravry, Victor
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Pikhurko, Oleg
    Mathematics Institute and DIMAP, University of Warwick, Coventry, United Kingdom.
    Vaughan, Emil
    London, Citymapper, United Kingdom.
    Volec, Jan
    Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Czech Republic.
    The codegree threshold of K-42023In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 107, no 5, p. 1660-1691Article in journal (Refereed)
    Abstract [en]

    The codegree threshold ex2 (n, F) of a 3-graph F is the minimum d = d(n) such that every 3-graph on n vertices in which every pair of vertices is contained in at least d + 1 edges contains a copy of F as a subgraph. We study ex2 (n, F) when F = K-4 , the 3-graph on 4 vertices with 3 edges. Using flag algebra techniques, we prove that if n is sufficiently large, then

    ex2 (n, K-4)⩽ (n + 1)/4.

    This settles in the affirmative a conjecture of Nagle [Congressus Numerantium, 1999, pp. 119-128]. In addition, we obtain a stability result: for every near-extremal configuration G, there is a quasirandom tournament T on the same vertex set such that G is o(n3)-close in the edit distance to the 3-graph C(T) whose edges are the cyclically oriented triangles from T. For infinitely many values of n, we are further able to determine ex2(n, K-4) exactly and to show that tournament-based constructions C(T) are extremal for those values of n.

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  • 3.
    Liu, Hong
    et al.
    Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea.
    Pikhurko, Oleg
    Mathematics Institute and DIMAP, University of Warwick, Coventry, United Kingdom.
    Sharifzadeh, Maryam
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Staden, Katherine
    School of Mathematics and Statistics, The Open University, Milton Keynes, United Kingdom.
    Stability from graph symmetrisation arguments with applications to inducibility2023In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 108, no 3, p. 1121-1162Article in journal (Refereed)
    Abstract [en]

    We present a sufficient condition for the stability property of extremal graph problems that can be solved via Zykov's symmetrisation. Our criterion is stated in terms of an analytic limit version of the problem. We show that, for example, it applies to the inducibility problem for an arbitrary complete bipartite graph B, which asks for the maximum number of induced copies of B in an n-vertex graph, and to the inducibility problem for K2,1,1,1 and K3,1,1, the only complete partite graphs on at most five vertices for which the problem was previously open.

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