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The model theory of unitriangular groups

Oleg V. Belegradek
1994 Annals of Pure and Applied Logic  
The model theory of groups of unitriangular matrices over rings is studied. An important tool in these studies is a new notion of a quasiunitriangular group.  ...  The models of the theory of all unitriangular groups (of fixed nilpotency class) are algebraically characterized; it turns out that all they are quasiunitriangular groups.  ...  In Section 3 we study the number of models in a power of the theory of a unitriangular group.  ... 
doi:10.1016/0168-0072(94)90022-1 fatcat:xynxr5q3pngnnmybuosai6ramq

Page 5259 of Mathematical Reviews Vol. , Issue 2001H [page]

2001 Mathematical Reviews  
Part 2 treats the model theory of unitriangular groups.  ...  (RS-KEME; Kemerovo) Model theory of unitriangular groups. Model theory and applications, 1-116, Amer. Math. Soc. Transl. Ser.2, 195, Amer. Math. Soc., Providence, RI, 1999.  ... 

Page 6412 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews  
The final section of the paper treats the number of models in a power of the theory of a unitriangular group. Many definitive results are obtained.  ...  (RS-KEME; Kemerovo) The model theory of unitriangular groups. (English summary) Ann. Pure Appl. Logic 68 (1994), no. 3, 225-261.  ... 

Quasiunitriangular groups

O. V. Belegradek
1993 Journal of Symbolic Logic (JSL)  
For a ring with unit R, which need not be associative, denote the group of upper unitriangular 3 × 3 matrices over R by UT3(R).  ...  Mal′cev [M] gave an algebraic characterization of the expanded groups of the form (R) as follows.  ...  Belegradek, "Some open problems in the model theory of unitriangular groups", Proceedings of the 10th Easter Conference on Model Theory, Humboldt Univ., Berlin, 1993, 9-13 Zentralblatt MATH [16]  ... 
doi:10.2307/2275333 fatcat:gpy4zta4ifcxjdcziendf2f2ce

On computable subgroups of the group of all unitriangular matrices over a ring

R.K. Tyulyubergenev
2017 BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS  
Groups of unitriangular matrices over the ring are a classic representative of the class of nilpotent groups and have numerous applications both in group theory and in its applications.  ...  On computable subgroups of the group of all unitriangular matrices over a ring The problems of existence and uniqueness of computable numberings are fundamental in theory of computably numbered groups.  ...  Our basic references for models, groups and rings are respectively [5] [6] [7] , of which we adopt terminology and notations.  ... 
doi:10.31489/2017m2/74-78 fatcat:nrk4khjlrrav5efdxbu63xe77q

Hopf monoids from class functions on unitriangular matrices

Marcelo Aguiar, Nantel Bergeron, Nathaniel Thiem
2013 Algebra & Number Theory  
We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal's category of species.  ...  Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids  ...  Acknowledgements We thank the referee and Franco Saliola for useful comments and suggestions.  ... 
doi:10.2140/ant.2013.7.1743 fatcat:ddv75cute5hzvewinu4wnsavwy

The polytabloid basis expands positively into the web basis [article]

Brendon Rhoades
2018 arXiv   pre-print
We show that the transition matrix from the polytabloid basis to the web basis of the irreducible S_2n-representation of shape (n,n) has nonnegative integer entries.  ...  This proves a conjecture of Russell and Tymoczko.  ...  The module W n is irreducible of shape (n, n) (see [5] ); the basis {w M : M ∈ W n } is the web basis of W n . The web basis is related to the invariant theory of the Lie group SL 2 .  ... 
arXiv:1808.00445v1 fatcat:unplvu6k2bhyxatwmmin456rhe

THE POLYTABLOID BASIS EXPANDS POSITIVELY INTO THE WEB BASIS

BRENDON RHOADES
2019 Forum of Mathematics, Sigma  
We show that the transition matrix from the polytabloid basis to the web basis of the irreducible $\mathfrak{S}_{2n}$ -representation of shape $(n,n)$ has nonnegative integer entries.  ...  This proves a conjecture of Russell and Tymoczko [Int. Math. Res. Not., 2019(5) (2019), 1479–1502].  ...  The author thanks an anonymous referee for their careful reading of this manuscript and numerous helpful suggestions for improving the exposition.  ... 
doi:10.1017/fms.2019.22 fatcat:f2q73h2nmbghrdrqfepy2lfvdq

Page 6174 of Mathematical Reviews Vol. , Issue 94k [page]

1994 Mathematical Reviews  
Unitriangular groups and undecidability. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 1993, no. 3, 19-22. The paper contains some strengthenings of the author’s earlier results.  ...  Here UT,,(R) denotes the group of all upper unitriangular n x n matrices over R. (2) For any Turing degrees d; and d> such that d, < d> there exists a ring R such that for any n > 3, Th(UT,,(R)) has degree  ... 

Page 1473 of Mathematical Reviews Vol. , Issue 2001C [page]

2001 Mathematical Reviews  
Then, via suitable interpretations, he shows that #(7) coincides with PC when T is the theory of the ring of integers, the theory of the field of rational numbers, or the theory of the group of 3 x 3 unitriangular  ...  As a corollary, it follows that #(T) = PC when T is the theory of rings, the theory of fields, or the theory of groups.  ... 

Page 01 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
Belegradek, Some open problems in the model theory of unitriangular groups (9-13); Sakaé Fuchino, Some problems of Séepin on openly generated Boolean algebras (14-29); D.  ...  In this very personal paper, the author gives his perspective on the development of set theory and model theory, and discusses some of his main sources of motivation.  ... 

A short proof on the transition matrix from the Specht basis to the Kazhdan-Lusztig basis [article]

Mee Seong Im
2019 arXiv   pre-print
We provide a short proof on the change-of-basis coefficients from the Specht basis to the Kazhdan-Lusztig basis, using Kazhdan-Lusztig theory for parabolic Hecke algebra.  ...  M.S.I. was partially supported by the Summer Collaborators Program at the School of Mathematics in the Institute for Advanced Study in Princeton, NJ.  ...  The author would like to thank Chun-Ju Lai, Arik Wilbert, and Jieru Zhu for insightful discussions.  ... 
arXiv:1912.03809v1 fatcat:md7i5gbtcfgq3ieqapbmgh3tre

Page 5284 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
Belegradek, Model theory of unitriangular groups (1-116); O. V. Belegradek, Model theory of locally free alge- bras [MR_ 89i:03056] (117-143); A. A.  ...  Kudaibergenov, The number of homogeneous models of a com- plete theory [MR 90f:03066] (187-204); T. G. Mustafin, The stability theory of polygons [MR 90b:03047] (205-223); E. A. Palyutin and S. S.  ... 

Page 5250 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews  
Let GL,,(F) be the group of all nonsingular n x n matrices over F and U,, the subgroup of upper unitriangular matrices in GL,,(F).  ...  The proof uses in an essential way the interplay between model theory of difference fields and Diophantine geometry, as developed by E. Hrushovski.  ... 

Page 6518 of Mathematical Reviews Vol. , Issue 93m [page]

1993 Mathematical Reviews  
The Mal’tsev correspondence is the functor R++ UT3(R) which associates with each ring R the nil-2 group UT3(R) of all upper unitriangular 3 x 3 matrices over R.  ...  In the present paper the author investigates the following main problem: To what extent do the isomorphism type of the mere group UT3(R) and the first-order theory of UT3(R) determine either R or the first  ... 
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