تقرير
Positivity and universal Pl\'ucker coordinates for spaces of quasi-exponentials
| العنوان: | Positivity and universal Pl\'ucker coordinates for spaces of quasi-exponentials |
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| المؤلفون: | Karp, Steven N., Mukhin, Evgeny, Tarasov, Vitaly |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematics - Complex Variables, Mathematical Physics, Mathematics - Representation Theory, 82B23, 15B48, 05E05, 14M15, 30C15 |
| الوصف: | A quasi-exponential is an entire function of the form $e^{cu}p(u)$, where $p(u)$ is a polynomial and $c \in \mathbb{C}$. Let $V = \langle e^{h_1u}p_1(u), \dots, e^{h_Nu}p_N(u) \rangle$ be a vector space with a basis of quasi-exponentials. We show that if $h_1, \dots, h_N$ are nonnegative and all of the complex zeros of the Wronskian $\operatorname{Wr}(V)$ are real, then $V$ is totally nonnegative in the sense that all of its Grassmann-Pl\"{u}cker coordinates defined by the Taylor expansion about $u=t$ are nonnegative, for any real $t$ greater than all of the zeros of $\operatorname{Wr}(V)$. Our proof proceeds by showing that the higher Gaudin Hamiltonians $T_\lambda^G(t)$ introduced in [ALTZ14] are universal Pl\"ucker coordinates about $u=t$ for the Wronski map on spaces of quasi-exponentials. The result that $V$ is totally nonnegative follows from the fact that $T_\lambda^G(t)$ is positive semidefinite, which we establish using partial traces. We also show that if $h_1 = \cdots = h_N = 0$ then $T_\lambda^G(t)$ equals $\beta^\lambda(t)$, which is the universal Pl\"ucker coordinate for the Wronski map on spaces of polynomials introduced in [KP23]. Comment: 24 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.20229 |
| رقم الأكسشن: | edsarx.2405.20229 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |