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Register a new userLittlewood-Richardson polynomials
بولينومات ليتلوود-ريتشاردسون
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Authors
A. I. Molev
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إننا نقدم عائلة من الحلقات من الدوال المتماثلة التي تعتمد على سلسلة غير محدودة من المعلمات. يتألف محور هذا الحلقة من المكائن المشابهة لدوال Schur. وتتألف معاملات الهيكل المرتبطة بها من المجموعات الثنائية في المعلمات التي نسميها Littlewood-Richardson polynomials. نقدم قاعدة تجميعية لحساب هذه المجموعات الثنائية باستخدام نتيجة سابقة من B. Sagan والمؤلف. يوفر القاعدة الجديدة نموذجا لهذه المجموعات الثنائية الذي يكون واضحا بشكل إيجابي في الإطار الذي أعطاه W. Graham. نطبق هذا النموذج لحساب ضرب الطبقات الشوبرت المتساوية على Grassmannians الذي يشير إلى خاصية الاستقرار لمعاملات الهيكل. كان النموذج الأول الذي يوفر نموذجا واضحا بشكل إيجابي للتوسع هذا من قبل A. Knutson وT. Tao باستخدام تجميع اللغز بينما لم يكن الإشارة إلى خاصية الاستقرار من هذا النموذج. نستخدم أيضا Littlewood-Richardson polynomials لوصف قاعدة الضرب في الجبر الخصوصي للجبر الخطي العام في الأساس الذي بناه A. Okounkov وG. Olshanski.
We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author. The new rule provides a formula for these polynomials which is manifestly positive in the sense of W. Graham. We apply this formula for the calculation of the product of equivariant Schubert classes on Grassmannians which implies a stability property of the structure coefficients. The first manifestly positive formula for such an expansion was given by A. Knutson and T. Tao by using combinatorics of puzzles while the stability property was not apparent from that formula. We also use the Littlewood-Richardson polynomials to describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by A. Okounkov and G. Olshanski.
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