Murphy Cellular Bases

• Murphy cellular bases are combinatorially explicit bases defined via Young tableaux and Jucys–Murphy elements that establish a cellular algebra framework.
• They enable the decomposition of representations by providing clear triangularity and filtration structures, which facilitate the computation of irreducible modules.
• These bases generalize to various algebras including q-Schur, Brauer, and Temperley–Lieb, supporting spectral analysis and effective induction and restriction.

Murphy cellular bases are distinguished, combinatorially explicit bases for (Iwahori-)Hecke algebras, symmetric group algebras, and a range of diagram algebras, which realize the algebra as a cellular algebra in the sense of Graham and Lehrer. These bases exploit the combinatorics of Young tableaux, the action of Jucys–Murphy elements (a maximal commutative subalgebra), and the structure of cell modules (Specht modules), leading to a powerful framework for decomposing representations, describing irreducibles, and computing decomposition numbers. The Murphy basis, originally defined for the symmetric groups, generalizes—sometimes via explicit algebraic or diagrammatic "lifting"—to Hecke algebras of complex reflection groups, q-Schur and Brauer algebras, partition algebras, BMW and Temperley–Lieb algebras, and their graded or cyclotomic variants. This cellular framework admits compatibility with restriction and induction, underlies the spectral theory of Jucys–Murphy elements, and encodes crucial combinatorial and order-theoretic information in its parametrization.

1. Core Construction and Definition

Murphy cellular bases arise from the synthesis of Young tableau combinatorics, Jucys–Murphy (JM) elements, and algebraic filtration by two-sided ideals. For the symmetric group algebra $\mathbf{k}[S_n]$ and its Hecke analogs, the construction is as follows (Grinberg, 28 Jul 2025, 1302.4272, 1101.2738, 1012.5983):

• Let $\lambda \vdash n$ be a partition of $n$. Consider the Specht module $S^\lambda$ indexed by standard Young tableaux of shape $\lambda$.
• The Jucys–Murphy elements are

$\mathbf{m}_k = \sum_{i=1}^{k-1} t_{i,k}$

where $t_{i,k}$ is the transposition $(i,k)$ in $S_n$. These elements commute, generating the Gelfand–Tsetlin algebra $GZ_n$.

• The Murphy basis for the group algebra (or Hecke algebra) is indexed by pairs of standard tableaux $(s,t)$ of the same shape $\lambda$ and can be realized via double coset representatives, or, more generally, as

$m_{s t} \in \mathbf{k}[S_n]$

such that $m_{st}^\dagger = m_{ts}$ under the algebra anti-involution, and (left or right) multiplication has an upper-triangular behavior with respect to dominance order on partitions.

For generalized $q$-Schur algebras $S$, the basis elements are constructed by "gluing" bases for the cell modules and their duals along defining idempotents (1012.5983): $C_{s,t}^\lambda = x_s \cdot 1_\lambda \cdot x_t^*$ where $x_s$ (resp.\ $x_t^*$) are basis elements for the cell module $A(\lambda)$ (resp.\ its dual under anti-involution $*$), and $1_\lambda$ is an orthogonal idempotent corresponding to weight $\lambda$.

These cellular bases satisfy the Graham–Lehrer axioms: parametrization by a poset, anti-involution swapping indices, and a triangularity condition for multiplication. The Murphy construction extends to a wide array of algebras, often using the combinatorics of paths in branching diagrams or diagrammatic calculus.

2. Cellular Structure: Filtration, Anti-involution, and Triangularity

Murphy bases are cellular in the sense of Graham–Lehrer, with a basis indexed by pairs in a finite partially ordered set (typically partitions), and structure constants that are triangular with respect to that order (1101.2738, 1302.4272). The structural features are:

• There exists an anti-involution $*$ (or $\dagger$) on the algebra satisfying $m_{st}^* = m_{ts}$.
• Multiplication by algebra elements acts "triangularly":

$$a \cdot m_{s t} \equiv \sum_{u} r_{su}(a) m_{u t} \pmod{A(>\lambda)}$$

where $A(>\lambda)$ is the two-sided ideal spanned by basis elements corresponding to shapes higher than $\lambda$ in dominance order, and $r_{su}(a)$ are scalars independent of $t$.

• This filtration and triangularity enable the construction of cell modules (Specht modules), whose irreducible quotients exhaust the simple modules in the semisimple case.

For generalized settings (e.g., $q$-Brauer algebras, partition algebras), the basis is built from appropriate combinatorial data (tableaux, Brauer diagrams, or one-column multipartitions) and the duality supplied by the anti-involution (1302.4272, Lobos et al., 2018, Mishra et al., 2019).

3. Role of Jucys–Murphy Elements and the Spectral Framework

The Jucys–Murphy elements generate a maximal commutative subalgebra (Gelfand–Tsetlin algebra) whose simultaneous diagonalization underpins the Murphy basis construction (Grinberg, 28 Jul 2025, Mishra et al., 2019, Rui et al., 2022, Lobos et al., 2018). Core properties:

• On any Specht module (classically), the collection $\{\mathbf{m}_1, ..., \mathbf{m}_n\}$ can be diagonalized with eigenvalues given by tableau "content" data.
• In cellular settings beyond type~A, these elements act "upper-triangularly" on the cellular basis, with combinatorially controlled diagonal entries. For example, in $q$-Brauer algebras, the Murphy basis is refined to a "JM-basis" where

$$m_{\mathfrak{t}} \cdot L_k = C_{\mathfrak{t}}(k) m_{\mathfrak{t}} + \sum_{s \succ t} a_{s,t} m_s$$

for combinatorially specified scalars $C_{\mathfrak{t}}(k)$ (Rui et al., 2022).

• The Jucys–Murphy spectral approach produces a canonical basis labeled by combinatorics of tableaux or Bratteli diagram paths, facilitating spectral decompositions, multiplicity-free towers, and explicit formulas for Gram determinants.

The spectral perspective is fundamental in the analysis of partition algebras for the rook monoid, where JM-elements uniquely label basis vectors in the Gelfand–Tsetlin basis, and their action is recursively specified across a tower of cellular algebras (Mishra et al., 2019).

4. Diagrammatic and Branching Generalizations: Towers, Jones Basic Construction, and Examples

Murphy cellular bases have been explicitly constructed for towers of algebras using Jones' basic construction, with the cellular structure and combinatorics propagated recursively (1106.5339). This framework generalizes as follows:

• Given a tower of cyclic cellular algebras $A_0 \subset A_1 \subset \cdots$, bases for each algebra can be indexed by pairs of paths in a branching diagram (Bratteli diagram).
• Basis elements take the form

$d_s^*\, c_{(\lambda, l)}\, d_t$

where $d_t$ is a product of "branching factors" associated to paths, $c_{(\lambda,l)}$ is a central idempotent, and $*$ denotes the involution.

• This method applies to Brauer, BMW, Temperley–Lieb, and partition algebras, yielding standard bases (sometimes coinciding with diagrammatic bases) that respect restriction and induction along the tower (1106.5339).
• For example, in the Temperley–Lieb and partition algebra cases, paths correspond to Dyck paths or set partitions, with Murphy-like bases coinciding with standard diagrammatic ones.

The construction respects the cellular order, coherently transfers cellular bases through restriction/induction, and allows the computation of semisimplicity criteria and Gram determinants via recursive techniques.

5. Graded and Cyclotomic Extensions; Categorification and Connections

Murphy bases have been generalized to graded and cyclotomic Hecke algebras, KLR (Khovanov–Lauda–Rouquier) algebras, and diagrammatic algebras, playing a critical role in modern categorification and modular representation theory (Bowman, 2017, Lobos et al., 2018):

• In cyclotomic (quiver) Hecke algebras, graded Murphy bases are constructed diagrammatically using box content orderings and explicit strand diagrams; these bases yield cellular structures that are compatible with Lusztig ${\bf a}_\sigma$-orderings.
• For generalized blob algebras, the graded cellular basis corresponds to elements

$$m_{\mathfrak{s},\mathfrak{t}} = \psi_{d(\mathfrak{s})}^*\, e(\mathbf{i}^\lambda)\, \psi_{d(\mathfrak{t})}$$

where $\psi_{d(\mathfrak{s})}$ are braid group generators and $e(\mathbf{i}^\lambda)$ encodes residue data from one-column multipartitions (Lobos et al., 2018).

These graded Murphy-type bases support categorification results such as Ariki's theorem, and their cell modules' decomposition matrices are shown to be unitriangular with respect to partial orders corresponding to crystal combinatorics and Kazhdan–Lusztig theory. The existence of multiple (Morita-equivalent) graded cellular bases is a further notable phenomenon (Bowman, 2017).

6. Applications: Representation Theory, Decomposition, and Semisimplicity Criteria

Murphy cellular bases provide a combinatorial and explicit toolkit for decomposing permutation and flag representation modules, computing irreducible constituents, and deriving semisimplicity criteria (1101.2738, 1302.4272, Rui et al., 2022):

• The basis allows the explicit construction of cell modules and determination of irreducible quotients, with multiplicities computable via combinatorial enumeration of tableau embeddings or orbits.
• Semisimplicity criteria for diagram algebras (e.g., $q$-Brauer) are deduced from the non-vanishing of Gram determinants associated to cell modules. For the $q$-Brauer algebra, semisimplicity occurs precisely when the quantum characteristic exceeds $n$ and parameter values $z^2 \neq q^{2a}$ for prescribed combinatorial exponents $a$ (Rui et al., 2022).
• Murphy bases facilitate the paper of modular and integral representations, determination of adjustment matrices, and explicit construction of irreducibles in categorification frameworks.

These bases are essential in centralizer algebra theory—such as for endomorphism algebras $\operatorname{End}_{U_q}(T)$ of $U_q$-tilting modules—where they relate the module structure of quantum groups to explicit cellular decompositions (Andersen et al., 2015).

7. Geometric and Flag-Theoretic Interpretation

The geometric underpinning of Murphy bases connects the orbits of flag varieties under classical groups (e.g., $GL_n(k)$) with the combinatorics of intersection matrices and the RSK correspondence (1101.2738):

• The orbits of the diagonal action on products of flags are encapsulated by intersection matrices, in bijection with pairs of semistandard tableaux.
• The Murphy basis arises as a characteristic function of such orbits, thus encoding not only algebraic but also geometric and combinatorial data.
• This perspective is leveraged in applications to representations over finite fields, modules over principal local rings, and connections with canonical bases in quantum group theory (1012.5983).

The interaction of algebraic, combinatorial, and geometric techniques in the construction, generalization, and application of Murphy cellular bases illustrates their central role in the contemporary representation theory of algebras.