Waldeck Schützer
Tenured Professor of Mathematics (Professor Associado IV)
In the Brazilian federal university system, Professor Associado IV is a permanent full professorship, not an "associate professor" position in the North American sense.
I teach and do research in the Department of Mathematics at the Universidade Federal de São Carlos (UFSCar), in São Carlos, Brazil.
My work is in algebra, mostly on the structure and symmetries of associative algebras: algebras graded by groups and their generalizations (H-comodule algebras), incidence algebras, and polynomial identities as a way to tell algebras apart. I like finding explicit descriptions, such as parametrizations, normal forms and criteria for when two algebras are the same, and principles for reducing a problem to simpler pieces and gluing them back together, in finite and infinite dimensions, including over indecomposable commutative rings. I also work in the representation theory of algebras and groups, and I enjoy its combinatorial and computational side.
I studied at ICMC-USP (BSc 1993, MSc 1996) and did my PhD at Rutgers University (2004) with Siddhartha Sahi, on combinatorial aspects of representation theory. I have been at UFSCar since 1996 and was Head of the Department of Mathematics from 2017 to 2019. Lately I have been computing polynomial identities of matrix algebras by machine, with each result certified by an independent bound. The outcome is an open database of identities, central polynomials, weak identities and A-identities in low degrees.
I write papers with colleagues and students. Outside research I take photographs and write creative code; my mathematical animations are posted as @infinitymathart. Questions and ideas are always welcome, so feel free to write.
What I am working on now
This is what I am working on at the moment.
- Certified computation of polynomial identities With H. L. Talpo I compute, by machine, the multiplicities of the irreducible representations of the symmetric group in the multilinear identities, central polynomials, weak identities and \(A\)-identities of the matrix algebras \(M_r(F)\), for \(r=2,\dots,6\) and degrees up to 31. Each value carries a status that says what is proved: exact, probable, or a bound. The results are in an open database, and the manuscript is in preparation. Open the database →
- Central polynomials of small degree The computations gave a central polynomial of degree 12 for \(M_4\), which refutes Formanek's conjectured minimum 13, and now one of degree 16 for \(M_5\), below the conjectured 19. They suggest a pattern that I conjecture holds for every \(r\): the shape \((2^{2r-2})\) occurs among the central polynomials of \(M_r\) in degree \(4(r-1)\), and that degree is the least possible. This is checked for \(r=2,3,4,5\) (the minimality only within the degrees stored). Proving it, or finding a uniform construction, is what I would like to do next. See the conjectures →
- Thresholds for positive multiplicities By work of Bergman and of Drensky and Kasparian, positivity of the multiplicities of the identities of \(M_r\) in all large degrees reduces to a single finite degree. The manuscript determines this threshold as \(5,9,14,21\) for \(r=2,3,4,5\) and bounds it between 27 and 30 for \(r=6\). In every proved case the last obstruction is a shape with two rows, and I conjecture that this always happens.
- Consequences of the standard polynomial A companion paper studies which identities of \(M_r\) in degrees \(2r+1\) to \(2r+4\) follow from the standard polynomial \(s_{2r}\). The computations show that all identities of degree 11 for \(M_4\) and of degree 13 for \(M_5\) are consequences of \(s_8\) and \(s_{10}\), respectively.
- Euler trails and leading cocharacters In a separate paper, I use Euler trails in graphs to describe the leading part of the cocharacters of matrix algebras, and show that the multiplicities of shapes with a fixed short tail are quasi-polynomials in the length of the first row, with explicit periods. It grew out of patterns first noticed in the computations. All publications →
- Gradings of infinite triangular matrices as non-associative algebras With F. Yasumura and Plamen Koshlukov, I am studying the group gradings of the algebra of infinite upper triangular matrices when it is regarded as a non-associative algebra. This continues my work with Yasumura on the associative case, where the gradings are classified up to graded isomorphism for an arbitrary group. Publications →
- Gradings on incidence algebras and graded bimodules With Felipe Hindi, I work on the group gradings of incidence algebras, including over commutative rings, and on the classification of graded bimodules over split group algebras, through stabilizers and Peirce decompositions. Two preprints on this are on arXiv. Publications →
- Teaching apps and mathematical art For my numerical calculus class I write small interactive apps that show the methods at work, in plain code. On the side I keep making animations and GPU particle shaders, posted as @infinitymathart. Apps →
Research interests
- Polynomial identities of matrix algebras
- H-comodule algebras, group gradings and graded identities
- Incidence algebras and their generalizations
- Hopf algebras and H-identities
- Representation theory: characters of Lie algebras and groups
- Algebraic combinatorics: Jack polynomials and Pieri rules
- Representations of the symmetric group and cocharacters
- Computer-assisted algebra with certified results
- Mathematical art, creative coding and photography
Name as it appears in databases: Schützer, Waldeck; Schutzer, W.; Schützer, W.
Positions
- 1996–Professor, Department of Mathematics, UFSCar (full-time, exclusive dedication). Rank: Professor Associado IV, effective February 10, 2025.
- 2015–2016Vice-Head of the Department of Mathematics, UFSCar
- 2017–2019Head of the Department of Mathematics, UFSCar
- 2019–2020Vice-Head of the Department of Mathematics, UFSCar
- 2019–2021Member of the Department Council, Department of Mathematics, UFSCar
- 2019–2021Elected member of the University Council (ConsUni), UFSCar
March 2019 to February 2021 - 2020Chair of the Electoral Commission for the succession of the UFSCar rectorship (CPE2020), by appointment
June 17 to December 7, 2020
Education
- 2004PhD in Mathematics, Rutgers, The State University of New Jersey (USA). Advisor: Siddhartha Sahi.
Thesis: On some combinatorial aspects of representation theory (CAPES fellowship). - 1996MSc in Mathematics, ICMC-USP. Advisor: Paulo Afonso Vieira da Veiga.
Dissertation: Lie algebras, Hopf algebras and quantum groups (CAPES fellowship). - 1993BSc in Mathematics, ICMC-USP (Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo).
Final project advisor: Marcos Nereu Arenales (cutting-stock problems; FAPESP fellowship).
Grants and projects
- 2021–Group gradings on infinite-dimensional triangularizable algebras
Classification of G-gradings (G arbitrary) of infinite upper triangular matrix algebras, up to graded isomorphism, with topological conditions. With Felipe Yukihide Yasumura. - 2020–The atomic decomposition of characters of semisimple Lie algebras
When do atomic numbers of an irreducible character come out non-negative? With Felipe de Mattos Chafik Hindi. - 2017–H-comodule algebras and their polynomial H-identities
Identifying H-comodule algebras that are distinguished by their H-identities. With Abel Gomes de Oliveira. - 2016–2020Computational study of low-degree polynomial identities of matrix algebras
Complete determination of identities, A-identities and central polynomials up to degree 13 for matrix algebras of orders 2 to 6. With Humberto Luiz Talpo. Completed; continued in the identities database. - 2004–2018Products of non-symmetric Jack polynomials
Combinatorial description of the expansion of a product of two non-symmetric Jack polynomials in the Jack basis. Completed.
Students
- 2026Felipe de Mattos Chafik Hindi, PhD, UFSCar. Existence conditions of atomic numbers and a classification theorem for graded incidence algebras.
- 2022Abel Gomes de Oliveira Júnior, PhD, UFSCar. H-comodule algebras and their polynomial H-identities.
- 2018Abel Gomes de Oliveira Júnior, MSc, UFSCar. Hopf algebras, Galois objects and polynomial identities.
- 2018Felipe de Mattos Chafik Hindi, MSc, UFSCar. Pieri formula for non-symmetric Jack polynomials.
- 2017Karina Branco da Cruz, MSc, UFSCar. Group representations and graded polynomial identities.
- 2015Gonzalo Emanuel Matías Gutiérrez, MSc, UFSCar. A study of the new character formula for representations of semisimple Lie algebras.
- 2014Marlon Pimenta Fonseca, MSc, UFSCar. Representations of the symmetric and alternating groups and applications to polynomial identities.
- 2012Alfredo Eduardo Calderón, MSc, UFSCar. Lie algebras and representation theory.
- 2004–2023Also 17 undergraduate final projects and 10 undergraduate research projects (iniciação científica) supervised.
Teaching
- 1996–Undergraduate (engineering, mathematics, chemistry): Calculus 1–3, Differential Calculus and Series, Numerical Calculus, Linear Algebra, Analytic Geometry, Algebraic Structures, Ring Theory, Introduction to Galois Theory, Real Numbers and Functions, Differential Equations.
- 2004–Graduate (PPGM, UFSCar): Algebra, Lie Algebras, Commutative Algebra, Introduction to Representation Theory, Representations of Hopf Algebras and Quantum Groups, Introduction to Symmetric Functions, Topics in Algebra.
- 2026Minicourse "Groups, Representations and the Algebra of the Rubik's Cube", IX Semana da Matemática da UFSCar (September 21–26). Permutation groups, group actions and representations, applied to the 3×3 Rubik's Cube and to a version where only the three central slices turn.
Talks
- 2023Graduações em álgebras de matrizes triangulares superiores de dimensão infinita (with F. Y. Yasumura)
- 2020Álgebras de incidência e uma introdução à combinatória algébrica
- 2018On the T-ideal generated by the standard polynomial of degree n
- 2018Sobre as consequências do polinômio standard (communication)
- 2015A-identidades da álgebra M2 sobre um corpo de característica zero (with H. L. Talpo and D. J. Gonçalves)
- 2014Álgebra com o cubo mágico
- 2006Towards the full Pieri rule for nonsymmetric Jack polynomials
Events organized
- 2026XXVIII Escola de Álgebra (Brazilian Algebra Meeting), ICMC-USP, São Carlos, June 29 to July 3: member of the local organizing committee
- 2015I Encontro de Álgebra e Topologia Algébrica da UFSCar (with D. J. Gonçalves, D. Vendruscolo, L. Hartmann and H. L. Talpo)
- 2010–2012Exam-application coordinator, OBMEP second phase
Examination committees
- 2017–2022PhD thesis committees: Abel Gomes de Oliveira Júnior (UFSCar, 2022), Dafne Campos Lima Bessades (UFMG, 2021), Mateus Eduardo Salomão (UFSCar, 2021), Dalton Couto Silva (UFSCar, 2020), Gabriel Lopes da Rocha (USP, 2017).
- 2009–2019Master's committees (UFSCar Physics, 2019; UNICAMP Electrical Engineering, 2009), four PhD qualifying exams (2020–2021), and many undergraduate final-project committees.
- 2010Public competition committee, Professor Adjunto, UFSCar
Service
- 2021–Referee, Journal of Mathematical Physics
- 2026–Referee, Latin American Journal of Mathematics
- 2018–Ad hoc reviewer, FAPESP (São Paulo Research Foundation): research projects and undergraduate, master's and doctoral fellowships
Research profiles
ORCIDORCID iD
0000-0001-9098-2519Currículo LattesarXivGoogle ScholarGoogle Scholar IDKELzmVsAAAAJMathSciNetMathSciNet author ID995806zbMATH OpenResearchGateGitHubElsewhere
Contact
Universidade Federal de São CarlosCentro de Ciências Exatas e de Tecnologia
Departamento de Matemática
Rod. Washington Luís, Km 235
13565-905 São Carlos, SP, Brasil
e-mail: waldeck@ufscar.br
Tel.: +55-16-3351-8221