Subroot systems and total positivity in finite reflection groups

Stempak, Krzysztof

doi:10.1090/proc/15979

Subroot systems and total positivity in finite reflection groups
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Proc. Amer. Math. Soc. 150 (2022), 4619-4627 Request permission

Abstract:

Given a root system $R$ and the corresponding finite reflection group $W$ let $\operatorname {Hom}(W,\,\widehat {\mathbb Z}_2)$ be the group of homomorphisms from $W$ into $\widehat {\mathbb Z}_2$, where $\widehat {\mathbb Z}_2=\{1,-1\}$ with multiplication. We propose a procedure of constructing subroot systems of $R$ by using homomorphisms $\eta \in \operatorname {Hom}(W,\,\widehat {\mathbb Z}_2)$. This construction is next used for establishing a relation between concepts of total positivity and $\eta$-total positivity.

References

Philippe Biane, Philippe Bougerol, and Neil O’Connell, Continuous crystal and Duistermaat-Heckman measure for Coxeter groups, Adv. Math. 221 (2009), no. 5, 1522–1583. MR 2522427, DOI 10.1016/j.aim.2009.02.016
Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 1827871, DOI 10.1017/CBO9780511565717
Kenneth I. Gross and Donald St. P. Richards, Total positivity, finite reflection groups, and a formula of Harish-Chandra, J. Approx. Theory 82 (1995), no. 1, 60–87. MR 1343132, DOI 10.1006/jath.1995.1068
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Richard Kane, Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 5, Springer-Verlag, New York, 2001. MR 1838580, DOI 10.1007/978-1-4757-3542-0
Jean B. Nganou, How rare are subgroups of index 2?, Math. Mag. 85 (2012), no. 3, 215–220. MR 2924161, DOI 10.4169/math.mag.85.3.215
Donald St. P. Richards and Kenneth I. Gross, Total positivity, harmonic analysis and random walks on Weyl chambers, Representation theory and harmonic analysis (Cincinnati, OH, 1994) Contemp. Math., vol. 191, Amer. Math. Soc., Providence, RI, 1995, pp. 153–161. MR 1365541, DOI 10.1090/conm/191/02329
Margit Rösler, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003), no. 6, 2413–2438. MR 1973996, DOI 10.1090/S0002-9947-03-03235-5
Krzysztof Stempak, Finite reflection groups and symmetric extensions of Laplacian, Studia Math. 261 (2021), no. 3, 241–267. MR 4321234, DOI 10.4064/sm200423-19-11