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Register a new userCharacters of the group $mathrm{EL}_d (R)$ for a commutative Noetherian ring $R$
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Let $R$ be a commutative Noetherian ring with unit. We classify the characters of the group $mathrm{EL}_d (R)$ provided that $d$ is greater than the stable range of the ring $R$. It follows that every character of $mathrm{EL}_d (R)$ is induced from a finite dimensional representation. Towards our main result we classify $mathrm{EL}_d (R)$-invariant probability measures on the Pontryagin dual group of $R^d$.
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Inner relations are derived between partial augmentations of certain elements (units or idempotents) in group rings.
We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings core graphs, and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>...$ such that $cap H_i={1}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.
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The branching theorem expresses irreducible character values for the symmetric group $S_n$ in terms of those for $S_{n-1}$, but it gives the values only at elements of $S_n$ having a fixed point. We extend the theorem by providing a recursion formula that handles the remaining cases. It expresses these character values in terms of values for $S_{n-1}$ together with values for $S_n$ that are already known in the recursive process. This provides an alternative to the Murnaghan-Nakayama formula.
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