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I. Gelʹfand, M. Kapranov, A. Zelevinsky (1994)
Discriminants, Resultants, and Multidimensional Determinants
In this paper we prove the case dim(V3)=3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$dim(V_3)=3$$\end{document} of a conjecture of the second author about the exterior operad ΛVdS2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\Lambda }^{S^2}_{V_d}$$\end{document}. For this we introduce a collection of natural involutions on the set of homogeneous cycle-free d-partitions of the complete graph K2d\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_{2d}$$\end{document}, and show that these involutions correspond to the relations in ΛVdS2(2d+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\Lambda }^{S^2}_{V_d}(2d+1)$$\end{document}. When d=3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d=3$$\end{document} this correspondence allows us to give an explicit description of a determinant-like map and to settle the above mentioned conjecture.
Monatshefte für Mathematik – Springer Journals
Published: Aug 1, 2022
Keywords: Exterior algebra; Edge-partitions of graphs; Determinants; Primary 15A15; Secondary 05C50; 05C70
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