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 Compositio MathematicaJournal Prestige (SJR): 3.139 Citation Impact (citeScore): 1Number of Followers: 1      Subscription journal ISSN (Print) 0010-437X - ISSN (Online) 1570-5846 Published by Cambridge University Press  [374 journals]
• ++++++++ ++++++++ ++++++++++++ ++++++++++++++++$\operatorname{SL}(n,\mathbb{Z})$ ++++++++++++ ++++++++ ++++&rft.title=Compositio+Mathematica&rft.issn=0010-437X&rft.date=2019&rft.volume=155&rft.spage=1245&rft.epage=1258&rft.aulast=Avni&rft.aufirst=Nir&rft.au=Nir+Avni&rft.au=Chen+Meiri&rft_id=info:doi/10.1112/S0010437X19007334">Words have bounded width in $\operatorname{SL}(n,\mathbb{Z})$
• Authors: Nir Avni; Chen Meiri
Pages: 1245 - 1258
Abstract: We prove two results about the width of words in $\operatorname{SL}_{n}(\mathbb{Z})$ . The first is that, for every $n\geqslant 3$ , there is a constant $C(n)$ such that the width of any word in $\operatorname{SL}_{n}(\mathbb{Z})$ is less than $C(n)$ . The second result is that, for any word $w$ , if $n$ is big enough, the width of $w$ in $\operatorname{SL}_{n}(\mathbb{Z})$ is at most 87.
PubDate: 2019-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X19007334
Issue No: Vol. 155, No. 7 (2019)

• The essential skeleton of a product of degenerations
• Authors: Morgan V. Brown; Enrica Mazzon
Pages: 1259 - 1300
Abstract: We study the problem of how the dual complex of the special fiber of a strict normal crossings degeneration $\mathscr{X}_{R}$ changes under products. We view the dual complex as a skeleton inside the Berkovich space associated to $X_{K}$ . Using the Kato fan, we define a skeleton $\text{Sk}(\mathscr{X}_{R})$ when the model $\mathscr{X}_{R}$ is log-regular. We show that if $\mathscr{X}_{R}$ and $\mathscr{Y}_{R}$ are log-smooth, and at least one is semistable, then $\text{Sk}(\mathscr{X}_{R}\times _{R}\mathscr{Y}_{R})\simeq \text{Sk}(\mathscr{X}_{R})\times \text{Sk}(\mathscr{Y}_{R})$ . The essential skeleton $\text{Sk}(X_{K})$ , defined by Mustaţă and Nicaise, is a birational invariant of $X_{K}$ and is independent of the choice of $R$ -model. We extend their definition to pairs, and show that if both
PubDate: 2019-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X19007346
Issue No: Vol. 155, No. 7 (2019)

• Affine cluster monomials are generalized minors
• Authors: Dylan Rupel; Salvatore Stella, Harold Williams
Pages: 1301 - 1326
Abstract: We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $\mathbf{g}$ -vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $A_{1}^{(1)}$ , we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.
PubDate: 2019-07-01T00:00:00.000Z
DOI: 10.1112/S0010437X19007292
Issue No: Vol. 155, No. 7 (2019)

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