| Summary: | We studied the Gaudin models with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">gl</mi><mo>(</mo><mn>1</mn><mo>|</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">gl</mi><mo>(</mo><mn>1</mn><mo>|</mo><mn>1</mn><mo>)</mo><mo>[</mo><mi>t</mi><mo>]</mo></mrow></semantics></math></inline-formula>-modules. Namely, we gave an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">gl</mi><mo>(</mo><mn>1</mn><mo>|</mo><mn>1</mn><mo>)</mo><mo>[</mo><mi>t</mi><mo>]</mo></mrow></semantics></math></inline-formula>-modules and showed that a bijection exists between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also gave dimensions of the generalized eigenspaces.
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