I work mainly in combinatorial representation theory. Specifically, I study how combinatorial objects (like partitions, tableaux, lattices, etc.) encode structure of algebras similar to the group algebra of the symmetric group, Brauer algebras, graded and affine Temperley-Lieb, Hecke, and Birman-Murakami-Wenzl algebras, etc.. My thesis work, completed in 2010 at the University of Wisconsin--Madison, began the study of two-boundary Hecke algebras.
The affine VW supercategory, with Martina Balagovic, Zajj Daugherty, Inna Entova-Aizenbud, Iva
Halacheva, Johanna Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova, and Catharina Stroppel, Selecta Math. (N.S.) 26 (2020), no. 2, #20, 42 pp.
Quasisymmetric Power Sums, with Cristina Ballantine, Angela Hicks, Sarah Mason, and Elizabeth Niese, J. Combin. Theory Ser. A 175 (2020); Extended abstract appearing in FPSAC 2018 Conference Proceedings, Sem. Lothar. Combin. 80B (2018) #25, 12 pp.
Abstract: In voting theory, simple questions can lead to complex and sometimes paradoxical results. Recently, we have been able to use tools from modern algebra to address long-standing arguments over voting and tallying methods that date back to the days of Jean-Charles de Borda and Nicolas de Condorcet. In this talk, we’ll explore how to frame voting methodology in terms of combinatorics and representations of finite groups.
Combinatorics and representation theory of diagram algebras., Colloquium, University of Victoria February 2020. [Slides (printable version)] [Abstract]
Abstract: In his early work, Schur constructed a powerful link between the symmetric group and the general linear group via commuting actions on a common vector space. Much more recently, there has been a cascade of developments surrounding diagram algebras that share commuting actions with other Lie groups, Lie algebras, and deformations thereof, with consequences in many fields, including representation theory, combinatorics, knot theory, and statistical mechanics. We will take a brief tour of some important examples of diagram algebras, and discuss how seeing them from Schur's perspective can reveal a wealth of information.
Representation theory and combinatorics of braid algebras and their quotients, Colloquium, UC Santa Barbara, January 2020. [Slides (.tex)] [Abstract]
Abstract: Classically, Schur-Weyl duality gave a beautiful interplay between the representation theory of the general linear groups and the symmetric groups, via their commuting actions on tensor spaces. Modernly, we see similar dualities giving rise to new interpretations of braid algebras and quotients thereof as commuting operators for many Lie algebras, quantum groups, and reflection groups. The structure that results gives us tools for interesting problems in representation theory, combinatorics, statistical mechanics, probability, and knot theory. In this talk, we will take a brief tour of some important examples from recent work.
Abstract: The signed Brauer algebra is a diagram algebra that arises as a centralizer algebra for the periplectic Lie super algebra, playing the role of the symmetric group in classical Schur-Weyl duality. The periplectic Lie super algebra p(n) is one of the so-called "strange" Lie superalgebras, and poses many challenges in its representation theory. In this talk, we will explore the "affine" or "graded" version of the signed Brauer algebra and its role in controlling the combinatorial representation theory of p(n).
Abstract: The classical, one-boundary, and two-boundary Temperley-Lieb algebras arise in mathematical physics related to solving certain rectangular lattice models. They also have beautiful presentations as “diagram algebras”, meaning that they have basis elements depicted as certain kinds of graphs, and multiplication rules are given by stacking diagrams and gluing of vertices. In this talk, we will explore these algebras and their representation theory, as well as their relationship to other important diagram algebras in combinatorial representation theory.
Abstract: Work of de Gier and Nichols explored the two-boundary Temperley-Lieb algebra as a natural generalization of the classical Temperley-Lieb algebra from a statistical mechanics perspective. They present the two-boundary Temperley-Lieb algebra both as a diagram algebra and as a quotient of the affine Hecke algebra of type C. In work with I. Halacheva, A. Ram, and A. Wilbert, we have expanded upon connections both to the two-poled braid group (via the type-C affine Hecke algebra) and to generalized exotic Springer fibers. In this talk, we will explore some of the beautiful representation theoretic and combinatorial structure of these algebras.
Abstract: In voting theory, simple questions can lead to complex and sometimes paradoxical results. Recently, we have been able to use tools from modern algebra to address long-standing arguments over voting and tallying methods that date back to the days of Jean-Charles de Borda and Nicolas de Condorcet. In this talk, we’ll explore how to frame voting methodology in terms of combinatorics and representations of finite groups.
Abstract: Representation theory is the study of abstract algebraic structures, like groups and vector spaces, by representing them as sets of matrices that obey the same addition and multiplication rules. This is powerful tool throughout mathematics, and also has important applications in physics, engineering, chemistry, and elsewhere. Combinatorial representation theory uses combinatorial tools, like partitions, tableaux, graphs, etc., to encode and keep track of representation theoretic data for many many wonderful groups, rings, algebras, etc. We'll briefly explore some examples of how this interplay between combinatorics and representations arises.
Abstract: In the 1995 paper entitled "Noncommutative symmetric functions," Gelfand et al. defined several noncommutative symmetric function analogues for well-known symmetric function bases, including two distinct types of power sum bases. This paper explores the combinatorial properties of their duals, two distinct quasisymmetric power sum bases. In particular, we show that they refine the classical symmetric power sum basis, and give transition matrices to other well-understood bases, as well as explicit formulas for products of quasisymmetric power sums. This is joint with with Cristina Ballantine, Angela Hicks, Sarah Mason, and Elizabeth Niese.
Abstract: The affine signed Brauer algebra arises in the study of the representation of the periplectic Lie super algebra via translation
functors, analogous to the role of the degenerate affine Hecke algebras in studying representations of the general linear
group. In this talk, we will explore its presentation as a diagram algebra, its natural action on tensor space, and other
combinatorial structure. This is joint work with Martina Balagovic, Inna Entova-Aizenbud, Iva Halacheva, Johanna
Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova, and Catharina Stroppel.
The ring of quasisymmetric functions QSym is a beautiful generalization of the classical ring of symmetric functions Sym, with many familiar bases having natural analogues. In particular, power sum symmetric functions play an important role in Sym�they satisfy many elegant combinatorial identities, and are instrumental in defining powerful inner products and homomorphisms on Sym. In this talk, I will discuss recent work building the corresponding quasisymmetric versions, and illustrate some of the parallel structure arising there. This is joint with with Cristina Ballantine, Angela Hicks, Sarah Mason, and Elizabeth Niese.
Abstract: The periplectic Lie superalgebra p(n) is one of the two so-called "strange" Lie superalgebras, in that it is not one of the analogs of the simple Lie algebras, nor is it of Cartan type. The representation theory of p(n) has posed a particular challenge �the category of finite dimensional representations has been shown to be a highest weight cate- gory, but otherwise early attempts at applying standard methods for classification had been unsuccessful. In this talk, we will explore some of the newly understood combinatorial structure of this category.
This is joint work with Martina Balagovic, Inna Entova-Aizenbud, Iva Halacheva, Johanna Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova, and Catharina Stroppel.
Abstract: In the study of symmetric functions, there are three important interconnected rings: Sym, (symmetric functions), NSym (noncommutative symmetric functions), and QSym (quasisymmetric functions). There are natural maps from both NSym and QSym to Sym, and a duality between NSym and QSym, all which provide a beautifully interconnected combinatorial and algebraic structure.
In Sym, power sum symmetric functions play an important role—they form a generating set, they appear as characters of associated algebras, and they satisfy many elegant combinatorial identities. When generalizing the power sums to NSym, two analogs were introduced by Gelfand et. al. via two natural choices of generating functions. In this talk, I will discuss recent work building the corresponding quasisymmetric versions, and illustrate some of the parallel structure arising there. This is joint with with Cristina Ballantine, Angela Hicks, Sarah Mason, and Elizabeth Niese.
Two-boundary diagram algebras, Colloquium, Bronx Community College, Bronx NY, October 2017. [Slides (.tex)] [Abstract]
Abstract: The classical two-boundary Temperley-Lieb algebra arises in mathematical physics related to lattice models. Much like the classical Temperley-Lieb algebra, it also has a beautiful presentation as a diagram algebra, with basis elements depicted as certain kinds of graphs that can be multiplied via gluing of vertices. In this talk, we will explore this algebra and its representation theory, as well as its relationship to other important diagram algebras in combinatorial representation theory. This is joint work with Arun Ram.
Representation theory and combinatorics of two-boundary Temperley-Lieb algebras, Mathematical Physics
Seminar, Centre de recherches mathematiques, Montreal, April 2017. [Abstract]
Abstract: Work of de Gier and Nichols explored the two-boundary Temperley-Lieb algebra as a natural generalization of the classical Temperley-Lieb algebra from a statistical mechanics perspective. They present the two-boundary Temperley-Lieb algebra both as a diagram algebra and as a quotient of the affine Hecke algebra of type C. In work with A. Ram, we have presented the affine Hecke algebra of type C as a centralizer algebra and as a quotient of the two-poled braid group. This work provides new tools for studying the two boundary Temperley-Lieb algebra, yielding beautiful combinatorial representation theoretic results.
Brauer algebras and their generalizations, Colloquium, New York City College of Technology,
Brooklyn, NY, March 2017. [Abstract]
Abstract: The Brauer algebra arises both combinatorially as an algebra spanned by certain diagrams (families of graphs), and algebraically in duality with orthogonal and symplectic groups. Through tinkering with the groups and algebras playing the role of the orthogonal and symplectic groups, or with the space through which we construct the duality, generalizations of the Brauer algebra appear that can also be defined through nice combinatorial rules. We will explore a few of these beautiful examples, particularly pertaining to Lie Groups and algebras, quantum groups, and Lie superalgebras.
Centralizers of the Lie superalgebra p(n) (where loops go to die), Representation Theory & Physics, University of Leeds, Leeds UK, July 2016. [Slides (.tex)] [Abstract]
Abstract: The Brauer algebra arises in Schur-Weyl duality with orthogonal and symplectic groups and Lie algebras.
Recently, D. Moon, followed by J. Kujawa and B. Tharp, studied a related algebra that centralizes
the action of the Lie superalgebra p(n), where sign changes appear in some relations and the parameter
associated to closed loops is set to 0. I will discuss these algebras, and explore how to construct
degenerate affine and cyclotomic versions. This is joint work with Martina Balagovic, Maria Gorelik,
Iva Halacheva, Johanna Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova, and
Catharina Stroppel, and a project begun at the BIRS workshop Women in Noncommutative Algebra
and Representation Theory
Representation theory and combinatorics of diagram algebras, 2nd Annual Graduate Student Conference in Algebra, Geometry, and Topology, Temple University, Philadelphia, PA, May 2016. [Slides (.tex)] [Abstract]
Abstract: Classical Schur-Weyl connects the representation theory of the general linear group to that of the symmetric
group via their commuting actions on a common tensor space. We will take a brief tour of other modern
examples of Schur-Weyl duality, and the consequent combinatorial results for algebras of braids, tangles and
other such families of diagrams.
Permutations, partitions, lattices, and some linear algebra: taste of combinatorial representation theory, 6th Annual York College Women In Mathematics Day, York College, New York, NY, April 2016. [Slides (.tex)] [Abstract]
Abstract: Representation theory is the study of abstract algebraic structures, like groups and vector spaces, by representing them as sets of matrices that obey the same addition and multiplication rules. By understanding the representations of a group, you can better understand the group itself, without having to deal with everything at once. Combinatorial representation theory (CRT) uses combinatorial tools (like partitions, tableaux, graphs, etc) to encode and keep track of the representation theory for many many wonderful groups, rings, algebras, etc.. We'll take a brief look at the poster child of CRT, the symmetric group, and see some examples of how combinatorics can be a powerful tool in studying algebraic objects.
Diagrammatics and actions of the affine BMW algebra, University of Melbourne, Australia, July 2015.
Representations of the two boundary Temperley-Lieb algebra, CUNY Representation Theory Seminar, New York, March 2015. [Slides] [Abstract]
Abstract: The classical Temperley-Lieb algebra arises both in
statistical mechanics and in Lie theory, in Schur-Weyl duality with
SL(2). The two-boundary Temperley- Lieb algebra arises as a
generalization of the classical case in the physical context, but also
in the algebraic context. In particular, it also presents in
Schur-Weyl duality with SL(2). Further, it presents as a diagram
algebra and as a quotient of the affine Hecke algebra of type C. I
will talk about how, in work with A. Ram, we have utilized these
algebraic presentations to provide new tools for studying the
two-boundary Temperley-Lieb algebra, yielding beautiful combinatorial
representation-theoretic results.
Representation theory and combinatorics of tensor power centralizer algebras, 8th Ausralia New Zealand Mathematics Convention, Melbourne, AU, December 2014. [Slides] [Abstract]
Abstract: Classical Schur-Weyl connects the representation theory of the general linear
group to that of the symmetric group via their commuting actions on a common
tensor space. We will see other modern examples of Schur-Weyl duality, and
explore how certain beautiful combinatorial statements about quantum groups,
tensor spaces, and corresponding diagram algebras arise.
Representations of the infinite symmetric group, Topology and Representation Theory at Kioloa, AU, November 2014. [Abstract]
Abstract: We review and introduce several approaches to the study of centralizer
algebras of the infinite symmetric group. Our work is
led by the double commutant relationship between finite symmetric
groups and partition algebras; in the case of the infinite symmetric group, we obtain
centralizer algebras that are contained in partition algebras. In
view of the theory of symmetric functions in non-commuting variables,
we consider representations of the infinite symmetric group that are faithful and
that contain invariant elements; namely, non-unitary representations
on sequence spaces.
Tensor spaces, braid groups, and some remarkable quotients, AGT Seminar, The University of Melbourne, September 2014. [Slides] [Abstract]
Abstract: In his thesis work, Schur constructed a powerful link between the symmetric group and the general linear group via commuting actions on a common vector space. More recently, there has been a cascade of developments surrounding centralizer algebras, with consequences in many fields including representation theory, combinatorics, knot theory, and statistical mechanics. We will take a brief tour of some important examples of centralizer algebras, and discuss how seeing them in this light can reveal a wealth of information.
Abstract: Work of de Gier and Nichols explored the two-boundary Temperley-Lieb algebra, which had arisen as a generalization of the classical Temperley-Lieb algebra via work in statistical mechanics. They present the two-boundary Temperley-Lieb algebra both as a diagram algebra and as a quotient of the affine Hecke algebra of type C. In work with A. Ram, we have presented the affine Hecke algebra of type C as a centralizer algebra and as a quotient of the two-poled braid group. This work provides new tools for studying the two boundary Temperley-Lieb algebra, yielding beautiful combinatorial representation theoretic results.
Abstract: The affine Birman-Murakami-Wenzl (BMW) algebras arise both as diagram algebras consisting of tangles and as algebras of operators commuting with the action of orthogonal and symplectic quantum groups on a particular tensor space. Their diagrammatic definitions rely on the choice of infinite families of complex parameters, a choice complicated by certain conditions studied at length in the literature. However, when viewing this algebra within the context of its action on tensor space, higher Casimir invariants inside the quantum group's centre present themselves as natural universal generalisations of complex parameters. In this talk, we will see how these universal parameters arise, and how their specialisation via an action on tensor space at once satisfies admissibility conditions and reflects the beautiful recursion relations arising in the literature. (Joint work with Arun Ram and Rahbar Virk.)
Representations of the infinite symmetric group, Poster session, FPSAC, Chicago IL, July 2014. [Poster] [Abstract]
Abstract: We review and introduce several approaches to the study of centralizer
algebras of the infinite symmetric group. Our work is
led by the double commutant relationship between finite symmetric
groups and partition algebras; in the case of the infinite symmetric group, we obtain
centralizer algebras that are contained in partition algebras. In
view of the theory of symmetric functions in non-commuting variables,
we consider representations of the infinite symmetric group that are faithful and
that contain invariant elements; namely, non-unitary representations
on sequence spaces.
Tensor spaces, braid groups, and some remarkable quotients, Colloquium, City College of NY, New York, NY, April 2014. [Slides] [Abstract]
Abstract: In his thesis work, Schur constructed a powerful link between the symmetric group and the general linear group via commuting actions on a common vector space. More recently, there has been a cascade of developments surrounding centralizer algebras, with consequences in many fields including representation theory, combinatorics, knot theory, and statistical mechanics. We will take a brief tour of some important examples of centralizer algebras, and discuss how seeing them in this light can reveal a wealth of information.
Tensor spaces, braid groups, and some remarkable quotients, Colloquium, University of Hawaii at Manoa, Honolulu HI, February 2014. [Slides] [Abstract]
Abstract: In his thesis work, Schur constructed a powerful link between the symmetric group and the general linear group via commuting actions on a common vector space. More recently, there has been a cascade of developments surrounding centralizer algebras, with consequences in many fields including representation theory, combinatorics, knot theory, and statistical mechanics. We will take a brief tour of some important examples of centralizer algebras, and discuss how seeing them in this light can reveal a wealth of information.
Centralizer Properties and Combinatorics of Affine Hecke Algebras of Type C, Lie Groups Seminar, Oklahoma State University, Stillwater OK, January 2014. [Slides]
Tensor spaces, braid groups, and some remarkable quotients, Colloquium, Oklahoma State University, Stillwater OK, January 2014. [Slides] [Abstract]
Abstract: In his thesis work, Schur constructed a powerful link between the symmetric group and the general linear group via commuting actions on a common vector space. More recently, there has been a cascade of developments surrounding centralizer algebras, with consequences in many fields including representation theory, combinatorics, knot theory, and statistical mechanics. We will take a brief tour of some important examples of centralizer algebras, and discuss how seeing them in this light can reveal a wealth of information.
Centralizers of the infinite symmetric group, Algebra seminar, University of Oregon, Eugene OR, December 2013. [Slides (.tex)] [Abstract]
Abstract: The partition algebra arises as the algebra of operators that commute with the diagonal action of the finite symmetric group S_n on the k-fold tensor product of the n-dimensional permutation module (akin to the classical Schur-Weyl duality between the symmetric group and the general linear group). Recently, this relationship between the symmetric group and the partition algebra has presented itself relevant to the study of several other algebraic topics, such as representation theoretic stability and the study of symmetric functions. In each of these contexts, however, a limit to the infinite symmetric group is taken, and a question arises as to what is the appropriate centralizer algebra. In this talk, I will explore several answers to this question, and how they depend on context. This work is joint with Peter Herbrich.
The quasi-partition algebra, Combinatorics seminar, UC Davis, Davis CA, May 2013. [Slides (.tex)] [Abstract]
Abstract: Several diagram algebras (like the group algebra of the symmetric group) arise via studying endomorphisms of tensor spaces that commute with other familiar groups or algebras (like the general linear group). The commutator relationships provide amazing tools for transferring representation theoretic information back and forth, and reveal beautiful combinatorial structure. In this talk, I will define the quasi-partition algebra, which arises as a centralizer algebra for the symmetric group. We will see how to recognize this algebra as a diagram algebra and explore some of its representation theory.
Combinatorics of affine Hecke algebras of type C, Combinatorics seminar, University of Washington, Seattle WA, May 2013. [Slides (.tex)] [Abstract]
Abstract: Several diagram algebras (like group algebras of symmetric groups or braid groups) arise as endomorphisms of tensor spaces that commute with classical Lie groups, Lie algebras, quantum groups, etc.. The commutator relationships provide amazing representation theoretic tools, and reveal beautiful combinatorial structure. In this talk I will describe how the affine type C Hecke algebra H arises as a diagram algebra. I will describe three ways of encoding irreducible representations of H--two classifications coming from the internal structure of the algebra (involving hyperplane and box arrangements) and one arising from its centralizer relationships (involving partitions and tableaux)--and show how to move between these encodings.
Combinatorics of the two-boundary Hecke algebra, Postdoc and graduate student seminar, Semester Program on "Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series", ICERM, April 2013. [Slides (.tex)]
The quasi-partition algebra, Discrete Math Day, WPI, Worcester MA, April 2013. [Slides (.tex)] [Abstract]
Abstract: Several diagram algebras (like the group algebra of the symmetric group) arise via studying endomorphisms of tensor spaces that commute with other familiar groups or algebras (like the general linear group). The commutator relationships provide amazing tools for transferring representation theoretic information back and forth, and reveal beautiful combinatorial structure. In this talk, I will define the quasi-partition algebra, which arises as a centralizer algebra for the symmetric group. We will see how to recognize this algebra as a diagram algebra and explore some of its representation theory.
Combinatorics of affine Hecke algebras of type C, Special Session on Combinatorial Avenues in Representation Theory, AMS meeting, Boulder CO, April 2013. [Slides (.tex)] [Abstract]
Abstract: Irreducible representations for the affine Hecke algebras of type C can be encoded combinatorially in a few different ways:
as points and regions in affine space relative to certain hyperplane arrangements, as arrangements and fillings of boxes with certain rotational symmetries, and now as certain partitions and tableaux arising from the representations of the complex Lie algebra gln. In this talk I will describe these three combinatorial models and the correspondence between
them.
The quasi-partition algebra, Discrete math seminar, Brown College, Providence RI, March 2013. [Slides (.tex)] [Abstract]
Abstract: Several diagram algebras (like the group algebra of the symmetric group) arise via studying endomorphisms of tensor spaces that commute with other familiar groups or algebras (like the general linear group). The commutator relationships provide amazing tools for transferring representation theoretic information back and forth, and reveal beautiful combinatorial structure. In this talk, I will define the quasi-partition algebra, which arises as a centralizer algebra for the symmetric group. We will see how to recognize this algebra as a diagram algebra and explore some of its representation theory.
A taste of combinatorial representation theory, Center for Women in Math at Smith College, March 2013. [Slides (.tex)] [Abstract]
Abstract: Representation theory is the study of abstract algebraic structures (e.g. groups) by representing them as sets of linear maps that obey the same addition and multiplication rules. By understanding the representations of a group, you can better understand the group itself, without having to deal with everything at once. Combinatorial representation theory (CRT) uses combinatorial tools (like partitions, tableaux, graphs, etc) to encode and keep track of the representation theory for many many wonderful groups, rings, algebras, etc.. We'll take a brief look at the poster child of CRT, the symmetric group, and see some examples of how combinatorics can be a powerful tool in studying algebraic objects.
10-min introduction, ICERM Semester Program on "Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series", January 2013. [Slides (.tex)]
Two boundary Hecke algebra and the affine Hecke algebra of type C, Special session on Algebraic Combinatorics and Representation Theory, Joint Math Meetings, January 2013. [Slides (.tex)] [Abstract]
Abstract: Several diagram algebras (like group algebras of symmetric groups or braid groups) arise as endomorphisms of tensor spaces that commute with classical Lie groups, Lie algebras, quantum groups, etc.. The commutator relationships provide amazing tools for studying the algebras' representation theory, and reveal beautiful combinatorial structure. This work
provides a bridge between results in quantum physics (and the two-boundary Temperley-Lieb algebra), the representation theory of the affine Hecke algebra of type C, and the combinatorics developed in thesis work on the degenerate two- boundary Hecke algebra, establishing a transfer of useful information between theses different points of view.This work is joint with Arun Ram.
The quasi-partition algebra, Algebra and combinatorics seminar, Loyola University, Chicago, December 2012. [Slides (.tex)] [Abstract]
Abstract: The partition algebra arises as a centralizer algebra for the symmetric group acting on the k-fold tensor product of its permutation representation. It also has a diagrammatic description, generated by set partitions of 2k elements (thus the name). The permutation representation is not in general irreducible, though. In this talk, I will define a new related algebra, the quasi-partition algebra, which also arises as a centralizer algebra for the symmetric group, but now acting on the k-fold tensor product of the large irreducible sub-module of the permutation representation. A similar diagrammatic description and some wonderful combinatorial results will be given. This work is joint with Rosa Orellana.
The quasi-partition algebra, Combinatorics seminar, University of Wisconsin, December 2012. [Slides] [Abstract]
Abstract: The partition algebra arises as a centralizer algebra for the symmetric group acting on the k-fold tensor product of its permutation representation. It also has a diagrammatic description, generated by set partitions of 2k elements (thus the name). The permutation representation is not in general irreducible, though. In this talk, I will define a new related algebra, the quasi-partition algebra, which also arises as a centralizer algebra for the symmetric group, but now acting on the k-fold tensor product of the large irreducible sub-module of the permutation representation. A similar diagrammatic description and some wonderful combinatorial results will be given. This work is joint with Rosa Orellana.
Universal parameters for centralizer algebras, Representation theory seminar, Northeastern University, April 2012. [Abstract]
Abstract: The affine Birman-Murakami-Wenzl (BMW) algebra arises are both as a algebra of tangles in a space with one puncture and as the centralizer of orthogonal and symplectic quantum groups on a certain tensor space of modules. The definition of the affine BMW algebra (and its degenerate version) relies on a choice of an infinite family of parameters, though this choice is not free; admissibility conditions have been studied at length. However, when viewing this algebra within the context of its action on tensor space, a family of central elements within the quantum group present themselves as a very natural choice of parameters. These central elements also arise in the study of higher Casimir invariants, and satisfy many beautiful recursion relations. I will outline how these elements arise in the study of centralizer algebras and how they fit into the larger study of both BMW algebras and Lie algebras and quantum groups. (Joint work with Arun Ram and Rahbar Virk.)
The two-boundary braid group and its amazing quotients, colloquium, Dartmouth College, March 2012. [Slides (.tex)] [Abstract]
Abstract: Around the turn of the 20th century, there was a wonderful breakthrough in studying infinite groups and their group-algebras: Schur-Weyl duality began as a transfer of information from the symmetric group to the general linear group via commuting actions on a very natural vector space. Since then, there has been an explosion of developments in tensor power centralizer algebras: algebras rising in duality with more familiar groups and algebras. Examples include the Brauer algebras, braid groups and quotients thereof, and tangle algebras. Sometimes we are so lucky as to discover that objects of outside interest also arise in duality with familiar algebras.
We'll take a brief tour of algebras arising as tensor power centralizer algebras, all of which will be generated by diagrams (like permutation diagrams, braids, tangles, etc.). Then we'll see how an algebra previously arising out of studies of reflection groups, a priori unrelated to centralizer algebras, is in fact a centralizer algebra itself. This is joint work with Arun Ram.
Type C symmetry of two-boundary Hecke algebras, AGC Seminar, San Francisco State University, March 2012. [Slides] [Abstract]
Abstract: Several diagram algebras (like group algebras of the symmetric groups or braid groups) arise via studying endomorphisms of tensor spaces that commute with classical Lie groups, Lie algebras, quantum groups, etc.. The commutator relationships provide amazing tools for studying the algebras' representation theory, and reveal beautiful combinatorial structure. In this talk, we will explore one case where we may be so lucky as to discover that certain already familiar algebras also carry commutator relationships. This is joint work with Arun Ram.
The affine BMW algebra and its degenerate version, Algebra, Geometry, and Topology Seminar, University of Melbourne, December 2011. [Abstract]
Abstract:
The BMW algebras arise both as diagram algebras and as algebras of operators which preserve symmetry in tensor products. I will define these algebras and explore some amazing properties which arise when studying central elements. This is joint work with Arun Ram and Rahbar Virk.
Type C symmetry in type A representation theory, Combinatorics Seminar, Dartmouth College,
November 2011.
Abstract:
The degenerate affine BMW algebra $W_k$ was introduced by Nazarov (there called the affine Wenzl algebra) in his study of the Brauer algebras via Jucys-Murphy elements. It arises in Schur-Weyl duality with the orthogonal and symplectic Lie algebras in the same way that the graded Hecke algebra of type A is in Schur-Weyl duality with the Lie algebras $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$. We will introduce the degenerate affine braid group, which has both the degenerate affine BMW algebra and the graded Hecke algebra as quotients. We will explore how results arise for $W_k$ which parallel those for the graded Hecke algebra.
Permutations, braids, partitions, and tableaux: finding the center of the affine BMW algebra, Research Seminar, St. Olaf Mathematics Dept, April 2011. [Abstract]
Abstract:
The "affine BMW algebra" developed out of a search for knot and link invariants. However, studying this algebra involves a lot more combinatorics than it does topology. In fact, most of the tools one needs to build its representations were developed for the purposes of understanding the symmetric group and its direct descendants. We'll pick a couple of these symmetric-group-like algebras, build some modules using partitions and Young tableaux, and use these big families of modules to deduce some structure about the algebras themselves.
Abstract:
The affine Birman-Murakami-Wenzl algebra arises both diagrammatically as an algebra of tangles in a punctured space, and algebraically via Schur-Weyl duality with the action of quantum groups of classical Lie type. Through exploring the structure of the affine BMW algebra and its degenerate version, we reveal that the center each algebra is generated by a nice family of symmetric functions. In particular the center of the degenerate affine BMW algebra is a ring that also arises in studying, for example, projective representation theory of the symmetric group, Chern numbers of kernel and cokernel bundles, and the homology of the loop space of the symplectic group.
Abstract: The degenerate two-boundary Hecke algebra arises as a centralizer algebra, much like the group algebra of the symmetric group, in duality with Lie algebras of type gl_n and sl_n. We will use combinatorics, symmetric functions, and braid groups to study the representation theory of this algebra and reveal a connection to the type C graded Hecke algebra.
Abstract: The affine Birman-Murakami-Wenzl algebra arises both diagrammatically as an algebra of tangles in a punctured space and algebraically via Schur-Weyl duality with the action of quantum groups of classical type. We will explore the structure of the affine BMW algebra and its degenerate version. In particular, we will take a look at the centers of these algebras, which are made up of nice families of symmetric functions. (Joint work with A. Ram and R. Virk)
The degenerate two-boundary Hecke algebra, Algebra, Geometry, and Topology Seminar, University of Melbourne, August 2010. [Abstract]
Abstract: In this talk, we will address a family of algebras similar to the group algebra of the symmetric group, the Brauer algebras, and the graded Hecke algebra of type A. In particular, we will discuss algebras of operators which commute with the action of Lie algebras
gl_n and sl_n on a specific tensor space. Combinatorial techniques
will help us explore the representation theory of these algebras,
revealing beautiful structure which mimics that of type C objects.
Two-boundary centralizer algebras, Dissertation defense, University of Wisconsin, May 2010.[Slides]
Degenerate two-boundary centralizer algebras, Algebraic Geometry Seminar, University of Iowa, April 16, 2010.[Abstract]
Abstract: Two-boundary centralizer algebras arise as algebras of commuting operators for a Lie algebra action on a tensor space. These algebras generalize the group algebra of the symmetric group, and graded Hecke and Brauer algebras, and have similarly elegant combinatorial structure. In this talk, we will explore tensor space of the form M \otimes N \otimes V^{\otimes k}. As an example, we will explore in detail the combinatorics of special cases corresponding to type A and explain how this might be applied to the study of the combinatorial representation theory of graded Hecke algebras of type C.
Abstract: We study algebras similar to the group algebra of the symmetric group, the Brauer algebras, and the graded Hecke algebra of type A. In particular, we investigate algebras of operators which commute with the action of $\mathfrak{sl}_n$ and $\mathfrak{gl}_n$ on tensor space of the form $M \otimes N \otimes L(\omega_1)^{\otimes k}$. We use combinatorial techniques to explore the structure and representation theory of these algebras, concentrating on cases where $M$ and $N$ are finite dimensional modules indexed by rectangular partitions. These examples yield beautiful structure and mimic that of type C objects.
Combinatorics of Centralizer Algebras, Colloquium, Swarthmore, April, 2010.
Combinatorics of Centralizer Algebras, Colloquium, Vassar College, March 30, 2010.
Algebra and Voting Theory, St. Olaf College, March 22, 2010.
Abstract: We study families of algebras that arise as algebras of commuting operators for the action of a finite dimensional complex reductive Lie algebra on a tensor space of the form $M \otimes N \otimes V^{\otimes k}$. This work uses similar techniques employed in the study of graded Hecke and Brauer algebras as centralizer algebras to construct two boundary analogs. In this talk, we outline this construction and explore some of the elegant combinatorial properties of the representation theory of specific examples.
Two boundary centralizer algebras, Lie theory seminar, University of Wisconsin -- Madison, October 5, 2009.
[Abstract]
Abstract: Two boundary diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the actions of Lie algebras on tensor spaces of the form $M \otimes N \otimes V^{\otimes k}$. As an example, we study in detail the combinatorics of special cases corresponding to gl_n and sl_n.
Combinatorics and the representation theory of centralizer algebras,
Pure maths student seminars, University of Melbourne, August 14, 2009.
[Abstract]
Abstract: There are many great examples of algebras which arise as tensor power centralizer algebras, algebras of operators which preserve symmetries in a tensor space. The most familiar example is the connection between the general linear group and the symmetric group, studied by Frobenius and Schur around 1900. The key is that these mutually centralizing algebras have linked representation theory, so the combinatorial tools used to study one algebra can be harnessed to study the other. In this talk, I will introduce some of these combinatorial tools, and talk about how to apply them to the representation theory of many favorite examples of diagram algebras.
Two boundary graded centralizer algebras, Algebra Seminar, University of Sydney, August 7, 2009.
[Abstract]
Abstract: Two boundary diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie group action on a tensor space, and show ties to already familiar objects. In particular, one two boundary centralizer algebra in type A is isomorphic to a graded Hecke algebra for a reflection group of type C. In this talk, I will construct this algebra, discuss its representations and their combinatorial structure, and provide its relation to type C objects.
Two boundary graded centralizer algebras, Tuesday seminar at Department of Mathematics and Statistics, University of Melbourne, August 4, 2009.
Building my favorite centralizer algebras, Combinatorics seminar, University of Wisconsin -- Madison, March 2009.
Abstract: The graded Birman-Murakami-Wenzl algebra arises as a tensor power centralizer algebra, an algebra of operators which preserve symmetries in a tensor space. The classical case, studied by Frobenius and Schur around 1900, provided the link between the representation theory of the symmetric group and the general linear group. In this talk, I will utilize combinatorial tools to explore the link between the actions of the graded BMW algebra and the symplectic and orthogonal Lie algebras on the tensor space.
Introduction to affine and graded BMW algebras, Representation Theory seminar, University of Wisconsin -- Madison, April 18, 2007.
Abstract: In voting theory, simple questions can lead to convoluted and sometimes paradoxical results. Recently, mathematician Donald Saari used geometric insights to study various voting schemes. He argued that a particular positional voting scheme (namely that proposed by Borda gives rise to the fewest paradoxes. In this talk, I presented an approach to similar ideas that draw from group theory and algebra. In particular, I employed tools from representation theory to elicit some of the natural behaviors of voting profiles. I also made generalizations to similar results for partially ranked data.
