### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 048, 28 pages      math.QA/0702277      https://doi.org/10.3842/SIGMA.2013.048
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

### Combinatorial Formulae for Nested Bethe Vectors

Vitaly Tarasov a, b and Alexander Varchenko c
a) Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA
b) St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia
c) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Received March 21, 2013, in final form June 27, 2013; Published online July 19, 2013

Abstract
We give combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for tensor products of irreducible evaluation modules over the Yangian $Y({\mathfrak{gl}}_N)$ and the quantum affine algebra $U_q(\widetilde{{\mathfrak{gl}}_N})$.

Key words: weight functions; nested Bethe vectors; algebraic Bethe ansatz.

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