We introduce the ramified partition algebra, which is a physically motivated and natural generalization of the partition algebra. We investigate its representation theory and demonstrate quasi--heredity under certain conditions. Under these conditions we determine an index set for simple modules and construct standard modules with this index set.
We construct tensor space representations of certain non--semisimple specializations, and show how to use these to build clock model transfer matrices in arbitrary physical dimensions.
In this paper we determine the complex generic representation theory of the Juyumaya algebra. We show that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by P. Martin.
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a can…
The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the algebra depend polynomially on a parameter. This is a survey paper which proves the primary results in the theory of partition algebras. Some of the results in this paper are new. This paper gives: (a) a presentation of the partition algebras by generators and relations, (b) shows that each partition algebra has an ideal which is isomorphic to a basic constructio…
We define an affine partition algebra by generators and relations and prove a variety of basic results regarding this new algebra analogous to those of other affine diagram algebras. In particular we show that it extends the Schur-Weyl duality between the symmetric group and the partition algebra. We also relate it to the affine partition category recently defined by J. Brundan and M. Vargas. Moreover, we show that this affine partition category is a full monoidal subcategory of the Heisenberg category.
We prove that the kernel of the action the group algebra of the Weyl group acting on tensor space (via restriction of the action from the general linear group) is a cell ideal with respect to the alternating Murphy basis. This provides an analogue of the second fundamental theory of invariant theory for the partition algebra over an arbitrary commutative ring and proves that the centraliser algebras of the partition algebra are cellular. We also prove similar results for the half partition algebras.
In this note we give representations for the partition algebra A_3(Q) in Young's seminormal form. For this purpose, we also give characterizations of A_n(Q) and$A_{n-1/2}(Q).
Let $V$ be a free module of rank $n$ over a commutative unital ring $k$. We prove that tensor space $V^{\otimes r}$ satisfies Schur--Weyl duality, regarded as a bimodule for the action of the group algebra of the Weyl group of $\mathrm{GL}(V)$ and the partition algebra $P_r(n)$ over $k$. We also prove a similar result for the half partition algebra.
In this paper, we realize the algebra of $\mathbb{Z}_2$-relations, signed partition algebras and partition algebras as tabular algebras and prove the cellularity of these algebras using the method of \cite{GM1}. Using the results of Graham and Lehrer in \cite{GL}, we give the modular representations of the algebra of $\mathbb{Z}_2$-relations, signed partition algebras and partition algebras.
We examine the structure of the partition algebra $P_n(δ)$ over a field $k$ of characteristic $p>0$. In particular, we describe the decomposition matrix of $P_n(δ)$ when $n<p$ and when $n=p$ and $δ=p-1$.
We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine for which exact values of the parameter the partition algebras are semisimple over an arbitrary field. Furthermore, we show that the blocks of the partition algebra over an arbitrary field categorify weight spaces of an action of type A quantum groups on an analogue of the Fock space. In particular, we recover the block structure which was recently determined by Bowman et al. In order to do so, we…
If $D$ is the definite quaternion algebra over $\qu$ of discriminant $p$, we compute, for any prime $p>3$, the number of infinite dimensional cusp forms on $D^*$ which are trivial at infinity, tamely ramified at $p$, and have given conductor $N$ away from $p$. We include a detail explanation of a Deuring--type correspondence between supersingular elliptic curves in characteristic $p$ and a certain double coset arising from the adelic points of $D^*$.
We study affine Grassmannians for ramified triality groups. These groups are of type ${}^3D_4$, so they are forms of the orthogonal or the spin groups in 8 variables. They can be given as automorphisms of certain twisted composition algebras obtained from the octonion algebra. Using these composition algebras, we give descriptions of the affine Grassmannians for these triality groups as functors classifying suitable lattices in a fixed space.
In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. In Part II, to the setup in Part I, we introduce Ome…
We prove a tamely ramified version of the Kazhdan-Lusztig equivalence using factorization algebras. More precisely, we establish an equivalence between the DG category of Iwahori-integrable affine Lie algebra representations and the DG category of representations of the "mixed" quantum group. This confirms a conjecture by D. Gaitsgory in arXiv:1810.09054 [math.RT].
Using a new presentation for partition algebras (J. Algebraic Combin. 37(3):401-454, 2013), we derive explicit combinatorial formulae for the seminormal representations of the partition algebras. These results generalise to the partition algebras the classical formulae given by Young for the symmetric group.
We introduce the quasi-partition algebra $QP_k(n)$ as a centralizer algebra of the symmetric group. This algebra is a subalgebra of the partition algebra and inherits many similar combinatorial properties. We construct a basis for $QP_k(n)$, give a formula for its dimension in terms of the Bell numbers, and describe a set of generators for $QP_k(n)$ as a complex algebra. In addition, we give the dimensions and indexing set of its irreducible representations. We also provide the Bratteli diagram for the tower of quasi-partition algebras (constructed by letting $k$ range over the positive intege…
We construct bases of the simple modules for partition algebras which are indexed by paths in an alcove geometry. This allows us to give a concrete interpretation (and new proof) of the monotone convergence property for Kronecker coefficients using stratifications of the cell modules of the partition algebra.
We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. We explain the limiting behavior and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition.
The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis ind…
We give a new presentation for the partition algebras. This presentation was discovered in the course of establishing an inductive formula for the partition algebra Jucys-Murphy elements defined by Halverson and Ram [European J. Combin. 26 (2005), 869-921]. Using Schur-Weyl duality we show that our recursive formula and the original definition of Jucys-Murphy elements given by Halverson and Ram are equivalent. The new presentation and inductive formula for the partition algebra Jucys-Murphy elements given in this paper are used to construct the seminormal representations for the partition alge…
We give combinatorial proofs of two identities from the representation theory of the partition algebra $C A_k(n), n \ge 2k$. The first is $n^k = \sum_λf^λm_k^λ$, where the sum is over partitions $λ$ of $n$, $f^λ$ is the number of standard tableaux of shape $λ$, and $m_k^λ$ is the number of "vacillating tableaux" of shape $λ$ and length $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is $B(2k) = \sum_λ(m_k^λ)^2$, where $B(2k)$ is the number of set partitions of $\{1, >..., 2k\}$. We show that this insertion restricts to work for th…
We use Naito-Sagaki's work [S. Naito & D. Sagaki, J. Algebra 245 (2001) 395--412, J. Algebra 251 (2002) 461--474] on Lakshmibai-Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type $A_{2\ell}^{(2)}$ and of type $D_{\ell+1}^{(2)}$. We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijectio…
This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(\mathbb{C}^n)^{\otimes k}$, where $\mathbb{C}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p =1$, the corresponding partition algebra was studied by Orellana. We also establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tan…
The restriction of a (dual) Specht module to a smaller symmetric group has a filtration by (dual) Specht modules of this smaller group. In the cellular structure of the group algebra of the symmetric group, the cell modules are exactly the (dual) Specht modules. The partition algebra is a cellular algebra containing the group algebra of the symmetric group. In this article, we study the structure of the restriction of a cell module to the group algebra of a symmetric group (with smaller index) and find conditions for the restriction to possess a (dual) Specht filtration.
We propose a new approach to study plethysm coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. This allows us to explain the stability properties of plethysm and Kronecker coefficients in a simple and uniform fashion for the first time. We prove the strengthened Foulkes' conjecture for stable plethysm coefficients in an elementary fashion.
We obtain several presentations by generators and relations for the rook partition monoids and algebras, as well as their singular ideals. Among other results, we also calculate the minimal sizes of generating sets (some of our presentations use such minimal-size generating sets), and show that the singular part of the rook partition monoid is generated by its idempotents.
We write down an explicit conjecture for the instanton partition functions in 4d N=2 SU(N) gauge theories in the presence of a certain type of surface operator. These surface operators are classified by partitions of N, and for each partition there is an associated partition function. For the partition N=N we recover the Nekrasov formalism, and when N=1+...+1 we reproduce the result of Feigin et. al. For the case N=1+(N-1) our expression is consistent with an alternative formulation in terms of a restricted SU(N)xSU(N) instanton partition function. When N=1+...+1+2 the partition functions can …
