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The Varchenko determinant of an oriented matroid | ||
| Transactions on Combinatorics | ||
| مقاله 2، دوره 10، شماره 4، اسفند 2021، صفحه 213-224 اصل مقاله (557.11 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2021.125990.1780 | ||
| نویسنده | ||
| Hery Randriamaro* | ||
| Lot II B 32 bis Faravohitra, 101 Antananarivo, Madagascar | ||
| چکیده | ||
| Varchenko introduced in 1993 a distance function on the chambers of a hyperplane arrangement that gave rise to a determinant whose entry in position $(C, D)$ is the distance between the chambers $C$ and $D$, and computed that determinant. In 2017, Aguiar and Mahajan provided a generalization of that distance function, and computed the corresponding determinant. This article extends their distance function to the topes of an oriented matroid, and computes the determinant thus defined. Oriented matroids have the nice property to be abstractions of some mathematical structures including hyperplane and sphere arrangements, polytopes, directed graphs, and even chirality in molecular chemistry. Independently and with another method, Hochst\"{a}ttler and Welker also computed in 2019 the same determinant. | ||
| کلیدواژه‌ها | ||
| Pseudohyperplane Arrangement؛ Distance؛ Determinant | ||
| مراجع | ||
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[1] M. Aguiar and S. Mahajan, Topics in Hyperplane Arrangements, Math. Surveys Monogr., 226 2017.
[2] A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler, Oriented Matroids, Encyclopedia Math. Appl., 46, Cambridge University Press, Cambridge, 1999. [3] R. Bland and M. Las Vergnas, Orientability of Matroids, J. Combin. Theory Ser. B, 24 (1978) 94–123.
[4] G. Denham and P. Hanlon, Some Algebraic Properties of the Schechtman-Varchenko Bilinear Forms, New Perspec- tives in Geometric Combinatorics, 38 MSRI Publications, 1999. [5] P. Deshpande, On Arrangements of Pseudohyperplanes, Proc. Indian Acad. Sci. Math. Sci., 126 (2016) 399–420.
[6] J. Folkman and J. Lawrence, Oriented Matroids, J. Combin. Theory Ser. B, 25 (1978) 199–236.
[7] Y. Gao and Y. Zhang, Diagonal Form of the Varchenko Matrices, J. Algebraic Combin., 48 (2018) 351–368.
[8] R. Gente, The Varchenko Matrix for Cones, PhD Thesis, Philipps-Universität Marburg, 2013.
[9] P. Hanlon and R. Stanley, A q-Deformation of a Trivial Symmetric Group Action, Trans. Amer. Math. Soc., 350 (1998) 4445–4459. [10] W. Hochstättler and V. Welker, The Varchenko Determinant for Oriented Matroids, Math. Z., 293 (2019) 1415–1430.
[11] A. Olzhabayev and Y. Zhang, Diagonal Form of the Varchenko Matrices for Oriented Matroids, arXiv: 2001.06460.
[12] H. Randriamaro, A Deformed Quon Algebra, Commun. Math., 27 (2019) 103–112.
[13] H. Randriamaro, A Multiparametric Quon Algebra, Bull. Iranian. Math. Soc., 47 (2021) 843–854.
[14] H. Randriamaro, The Varchenko Determinant for Apartments, Results Math., 75 (2020) pp. 17.
[15] V. Schechtman and A. Varchenko, Quantum Groups and Homology of Local Systems, Algebraic geometry and analytic geometry (Tokyo, 1990), 182–197, Satell. Conf. Proc., Springer, Tokyo, 1991. [16] A. Varchenko, Bilinear Form of Real Configuration of Hyperplanes, Adv. Math., 97 (1993) 110–144.
[17] G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 1995. | ||
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