Masashi Kosuda
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).
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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.
In this paper we describe the blocks of the partition algebra over a field of positive characteristic.
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.
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.
Semisimplicity criterion for 2-tonal partition algebras
本文研究2音调划分代数的半单性条件，证明当参数δ不在非负整数集时，所有n的偶划分代数P_n^2(δ)均为半单。
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 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.
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…
In this paper we show that the center of the partition algebra $\mathcal{A}_{2k}(δ)$, in the semisimple case, is given by the subalgebra of supersymmetric polynomials in the normalised Jucys-Murphy elements. For the non-semisimple case, such a subalgebra is shown to be central, and in particular it is large enough to recognise the block structure of $\mathcal{A}_{2k}(δ)$. This allows one to give an alternative description for when two simple $\mathcal{A}_{2k}(δ)$-modules belong to the same block.
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.
For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[δ]$. We construct modules $\{ Δ_{\underlineμ} \}_{\underlineμ}$ for $P^l_n$ over $\mathbb{Z}[δ]$, and hence over any integral domain containing $\mathbb{Z}[δ]$ that is a $\mathbb{Z}[δ]$-algebra (such as $\mathbb{C}[δ]$), that pass to a complete set of irreducible modules over the field of fractions. We show that $P^l_n$ is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer [6]. (The aim is to investigate the non-semisimple structure of the tonal part…
In this thesis, we investigate the asymptotics of random partitions chosen according to probability measures coming from the representation theory of the symmetric groups $S_n$ and of the finite Chevalley groups $GL(n,F_q)$ and $Sp(2n,F_q)$. More precisely, we prove laws of large numbers and central limit theorems for the $q$-Plancherel measures of type A and B, the Schur-Weyl measures and the Gelfand measures. Using the RSK algorithm, it also gives results on longest increasing subsequences in random words. We develop a technique of moments (and cumulants) for random partitions, thereby using…
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.
The main result here gives an algebra(/linear category) isomorphism between a geometrically defined subcategory $J^1_0$ of a short Brauer category $J_0$ and a certain one-parameter specialisation of the blob category $b$. That is, we prove the Conjecture in Remark 6.7 of [14]. We also define a sequence of generalisations $J^i_{i-1}$ of the category $J^1_0$. The connection of $J_0$ with the blob category inspires a search for connections also with its beautiful representation theory. Here we obtain formulae determining the non-semisimplicity condition (generalising the classical `root-of-unity'…
Character sheaves on loop Lie algebras: polar partition
受J-K. Yu超尖点表示构造启发，研究提出由极化数据参数化的loop李代数极化划分，引出其特征层的猜想性构造。
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…
A diagram-like basis for the multiset partition algebra
本文为多重集分拆代数构造了类图基，并利用该基给出不可约表示和生成集，同时建立了从轨道基到类图基的基变换，揭示了其与分拆代数结构的相似性。
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 introduce multi-colour partition algebras $P_{n,m}(δ_0, ..., δ_{m-1})$, which are generalization of both bubble algebras and partition algebras, then define the bubble algebra $T_{n,m}(δ_0, ..., δ_{m-1})$ as a sub-algebra of the algebra $P_{n,m}(δ_0, ..., δ_{m-1})$. We present general techniques to determine the structure of the bubble algebra over the complex field in the non-semisimple case.
Permutation modules play an important role in the representation theory of the symmetric group. Hartmann and Paget defined permutation modules for non-degenerate Brauer algebras. We generalise their construction to a wider class of algebras, namely cellularly stratified algebras, satisfying certain conditions. Partition algebras are shown to satisfy these conditions, provided the characteristic of the underlying field is large enough. Thus we obtain a definition of permutation modules for partition algebras.
Iwahori-Hecke algebras acting on tensor space by $q$-deformed letter permutations and $q$-partition algebras
本研究构造了Iwahori-Hecke代数在张量空间上的$q$-形变作用，定义了对应的$q$-分划代数，并证明其与Halverson和Thiem此前定义的$q$-分划代数同构。
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…
For a cellular algebra $\A$ with a cellular basis $\ZC$, we consider a decomposition of the unit element $1_\A$ into orthogonal idempotents (not necessary primitive) satisfying some conditions. By using this decomposition, the cellular basis $\ZC$ can be partitioned into some pieces with good properties. Then by using a certain map $\a$, we give a coarse partition of $\ZC$ whose refinement is the original partition. We construct a Levi type subalgebra $\aA$ of $\A$ and its quotient algebra $\oA$, and also construct a parabolic type subalgebra $\tA$ of $\A$, which contains $\aA$ with respect to…
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.
Branching rules for cell modules on a tower of the partition algebras
本文利用图式化方法研究分拆代数塔上胞模的分支法则，并据此计算其格罗滕迪克环的结构常数。
Kudryavtseva and Mazorchuk exhibited Schur-Weyl duality between the rook monoid algebra $\mathbb{C}R_n$ and the subalgebra $\mathbb{C}I_k$ of the partition algebra $\mathbb{C} A_k(n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. In this paper, we consider a subalgebra $\mathbb{C}I_{k+\frac{1}{2}}$ of $\mathbb{C} I_{k+1}$ such that there is Schur-Weyl duality between the actions of $\mathbb{C} R_{n-1}$ and $\mathbb{C} I_{k+\frac{1}{2}}$ on $(\mathbb{C}^n)^{\otimes k}$. This paper studies the representation theory of partition algebras $\mathbb{C}I_k$ and $\mathbb{C}I_{k+\frac{1}{2}}$ for rook monoid…
In this paper we shall investigate further the connections between the postprojective partition of an algebra and its Auslander-Reiten quiver.
Kostant's partition function counts the number of ways to represent a particular vector (weight) as a nonnegative integral sum of positive roots of a Lie algebra. For a given weight the $q$-analog of Kostant's partition function is a polynomial where the coefficient of $q^k$ is the number of ways the weight can be written as a nonnegative integral sum of exactly $k$ positive roots. In this paper we determine generating functions for the $q$-analog of Kostant's partition function when the weight in question is the highest root of the classical Lie algebras of types $B$, $C$ and $D$.
Super Multiset RSK and a Mixed Multiset Partition Algebra
本文将分拆代数与多重集分拆代数的表示对偶性推广到外幂，引入混合多重集分拆代数，并推广 RSK 算法以得到计数结果，进而描述交换与反交换变量多项式环在一般线性群和对称群下的模分解。
Representations of the quasi-partition algebras
本文推广拟划分代数的定义并构造其单模，同时引入两个新代数，通过广义 Bratteli 图揭示三者表示论的联系，并给出单模维数的组合公式。
We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence of length $n-r$ index a subset of the set of all $r$th Kronecker powers of $n \times n$ permutation matrices which is a basis for the linear span of that set. Thanks to a known Schur--Weyl duality, this gives a new basis for the centralizer algebra of the partition algebra acting on the $r$th tensor power of a vector space. We give some related results on the set of doubly stochastic matrices in that algebra.
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 …
Half-Diagrams in Partition Algebras: A Geometric Perspective on Multiplicities
本文利用 Bowman-De Visscher-Orellana 公式计算分区代数中半图模的限制重数，并通过平面三角形与圆锥曲线给出几何解释，揭示几何与代数联系的内在原因。
Permutation invariance, partition algebras and large $N$ matrix models
本文研究离散规范群 $S_N$ 的矩阵模型，利用广义 Schur-Weyl 对偶引入划分代数，计算 $S_N$ 不变可观测量期望值，并探讨其在矩阵量子力学中的应用。
The homology of the partition algebras
本文研究了划分代数的同调，证明其与对称群同调在随节点数增加的度数范围内同构，并在参数 $δ$ 可逆时在所有度数上同构，丰富了代数同调稳定性文献。
This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition fu…
Stable homology isomorphisms for the partition and Jones annular algebras
本文证明琼斯环形代数同调在梯度1/2直线以下与循环群同调同构，分划代数同调在梯度1直线以下与对称群同调同构（加强已有结果），且同构区间超代数稳定范围，还得琼斯环形代数奇链与可逆参数结果。
We compute the representation theory of two families of noncrossing partition quantum groups connected to amalgamated free products and free wreath products. This illustrates the efficiency of the methods developed in our previous joint work with M. Weber.
In this paper, we introduce Gram matrices for the signed partition algebras, the algebra of $\mathbb{Z}_2$-relations and the partition algebras. We prove that the Gram matrix is similar to a matrix which is a direct sum of block submatrices. In this connection, $(s_1, s_2, r_1, r_2, p_1, p_2)$-Stirling numbers of the second kind are introduced and their identities are established. As a consequence, the semisimplicity of a signed partition algebra is established.
The partition algebra $\mathsf{P}_k(n)$ and the symmetric group $\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$, so there is a surjection $\mathsf{P}_k(n) \to \mathsf{Z}_k(n) := \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities …
Cellular subalgebras of the partition algebra
本文研究分区代数等图代数的细胞结构，利用半图分解证明相关图代数均为细胞代数并参数化其细胞模，同时提出从一般细胞代数与 rook Brauer 代数子代数构造新细胞代数的 cellular wreath product 方法。
Semi-simple partition algebras as centralizers of representations for rook monoids
本文证明了半单划分代数可作为车 monoid 平凡表示的迭代限制-诱导表示的中心化子代数，并给出了该表示到张量空间的分解及由广义 Bell 数给出的维数。
We introduce a new basis of the Temperley-Lieb algebra. It is defined using a bijection between noncrossing partitions and fully commutative elements together with a basis introduced by Zinno, which is obtained by mapping the simple elements of the Birman-Ko-Lee braid monoid to the Temperley-Lieb algebra. The combinatorics of the new basis involve the Bruhat order restricted to noncrossing partitions. As an application we can derive properties of the coefficients of the base change matrix between Zinno's basis and the well-known diagram or Kazhdan-Lusztig basis of the Temperley-Lieb algebra. I…
The symmetric group $\mathsf{S}_n$ and the partition algebra $\mathsf{P}_k(n)$ centralize one another in their actions on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the $n$-dimensional permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$. The duality afforded by the commuting actions determines an algebra homomorphism $Φ_{k,n}: \mathsf{P}_k(n) \to \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$ from the partition algebra to the centralizer algebra $\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$, which is a surjection for all $k, n \in \mathbb{Z}_{\ge 1}$, and an isom…
We compute the dimension $d_{n,r}(q) = \dim(\IR_q^r)$ of the defining module $\IR_q^r$ for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on $\GL_n(\FF_q)$. This dimension is a polynomial in $q$ that specializes as $d_{n,r}(1) = n^r$ and $d_{n,r}(0) = B(r)$, the $r$th Bell number. We compute $d_{n,r}(q)$ in two ways. The first is purely combinatorial. We show that $d_{n,r}(q) = \sum_λf^λ(q) m_r^λ$, where $f^λ(q)$ is the $q$-hook number and $m_r^λ$ is the number of $r$-vacillating tableaux. Using a Schensted bijection, we write this …
The partition algebra and the plethysm coefficients II: ramified plethysm
本文利用对称群圈积与分歧分划代数的Schur--Weyl对偶，将plethysm系数解释为模限制的合成因子重数，揭示其首部分较大时的稳定现象，给出μ为单部分时的首个正公式及两类系数的显式公式与组合解释。
The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a partition $λ$ is irreducible. This is done by extending the results of James and Mathas. These descriptions depend on the crystal of the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to determine which irreducible modules have a realization as a Specht module. To do this, a new condition of irr…
Subdivision of KLRW Algebras in Affine Type A
本文研究仿射A型KLRW代数间的细分映射，证明分拆索引幂等元的像仍为分拆索引幂等元，给出新分拆形式，并建立某些分次分解数的等式。
We classify extremal traces on the seven direct limit algebras of noncrossing partitions arising from the classification of free partition quantum groups of Banica-Speicher (arXiv:0808.2628) and Weber (arXiv:1201.4723). For the infinite-dimensional Temperley-Lieb-algebra (corresponding to the quantum group $O^+_N$) and the Motzkin algebra ($B^+_N$), the classification of extremal traces implies a classification result for well-known types of central random lattice paths. For the $2$-Fuss-Catalan algebra ($H_N^+$) we solve the classification problem by computing the \emph{minimal or exit bounda…
In this paper we conjecture combinatorial Rogers-Ramanujan type colored partition identities related to standard representations of the affine Lie algebra of type $C^{(1)}_\ell$, $\ell\geq2$, and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.
We give closed formulae for the q-characters of the fundamental representations of the quantum loop algebra of a classical Lie algebra in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of the eigenvalues of the imaginary subalgebra in terms of these partitions.
Assume $\mathsf{M}_n$ is the $n$-dimensional permutation module for the symmetric group $\mathsf{S}_n$, and let $\mathsf{M}_n^{\otimes k}$ be its $k$-fold tensor power. The partition algebra $\mathsf{P}_k(n)$ maps surjectively onto the centralizer algebra $\mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$ for all $k, n \in \mathbb{Z}_{\ge 1}$ and isomorphically when $n \ge 2k$. We describe the image of the surjection $Φ_{k,n}:\mathsf{P}_k(n) \to \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k})$ explicitly in terms of the orbit basis of $\mathsf{P}_k(n)$ and show that when $2k > n$ th…
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