Seminormal Forms in Algebra & Geometry

• Seminormal forms are canonical methods in algebraic geometry and representation theory that simplify diagonalization and bridge singular and regular structures.
• In commutative algebra, seminormalization offers a precise approach to achieving near-normality by preventing proper subintegral extensions while maintaining topological and field properties.
• In representation theory, seminormal bases leveraging Jucys–Murphy elements provide explicit idempotents and effective computational tools for module decomposition and structural analysis.

A seminormal form is a canonical diagonalization or normalization procedure arising in both algebraic geometry and in the representation theory of finite and diagrammatic algebras. Seminormality in commutative algebra and algebraic geometry distinguishes a class of reduced schemes that admit no proper subintegral extensions, providing a universal object “closest” to normality without being integrally closed. In algebra and combinatorics, seminormal forms refer to bases and idempotents that diagonalize a maximal commutative family (often the Jucys–Murphy elements) in representations of symmetric groups, Hecke algebras, and related diagrammatic algebras, leading to explicit and computationally efficient descriptions of modules, branching rules, and Gram determinants.

1. Seminormality in Commutative Algebra and Algebraic Geometry

A commutative reduced ring $A$ is seminormal if, whenever $b^3 = c^2$ in $A$, there exists $a \in A$ such that $a^2 = b$ and $a^3 = c$. Equivalently, $A$ admits no proper subintegral extension inside its total ring of fractions; a subintegral extension $A \subset B$ is one inducing a homeomorphism on prime spectra and isomorphisms on all residue fields. The seminormalization $A^{\mathrm{sn}}$ is the universal subintegral extension of $A$.

For a reduced finite-type scheme $b^3 = c^2$0 over a Nagata base, the seminormalization $b^3 = c^2$1 is a proper morphism inducing a bijection on the underlying topological spaces and isomorphisms on all residue fields; universality is characterized by the property that any morphism from a seminormal $b^3 = c^2$2 factors uniquely through $b^3 = c^2$3 (Huber et al., 2017). Kollár extended this concept by defining seminormality for pairs $b^3 = c^2$4, with $b^3 = c^2$5 a closed, nowhere dense subset, resulting in notions local in both the Zariski and adic topologies and stable under completion and regular base change (Kollár, 2015).

The seminormal locus is intrinsic: the category of seminormal schemes is equivalent (over a Nagata base) to the category of representable cdh-sheaves, with $b^3 = c^2$6 and inverse given by recovering the underlying seminormal scheme from the sheaf. Over real algebraic varieties, seminormalization may be refined relative to the "central locus," reflecting the topological nature of real points and hereditarily rational functions (Fichou et al., 2017).

2. Seminormal Forms in Finite and Diagram Algebras

In representation theory, a seminormal form describes a basis for a module such that a maximal commutative subalgebra—typically generated by Jucys–Murphy elements—is simultaneously diagonal. The canonical example is Young’s seminormal form for Specht modules of the symmetric group: constructing for each standard tableau $b^3 = c^2$7 a primitive idempotent $b^3 = c^2$8 in $b^3 = c^2$9 such that $A$0, with $A$1 the content of $A$2 (Ryom-Hansen, 2011, Ryom-Hansen, 2013).

Generalizations have been developed for cyclotomic and affine Hecke algebras, quiver Hecke (KLR) algebras, partition algebras, and the Temperley–Lieb algebra, as well as the super versions such as cyclotomic Hecke–Clifford algebras (Hu et al., 2013, Li et al., 21 Feb 2025, Hu et al., 11 Dec 2025, Bastías et al., 2023, Enyang, 2011). For each, a family of commuting Jucys–Murphy elements gives rise to explicit seminormal idempotents and bases, allowing for the diagonalization of key operators and often controlling the fine structure of the associated cell or Specht modules.

3. Construction and Universal Properties of Seminormalization

For a reduced Noetherian ring $A$3 with normalization $A$4, the absolute seminormalization is given by

$A$5

in other words, elements in the normalization that induce elements in all residue fields of $A$6 (Kollár, 2015). For schemes, the seminormalization $A$7 is characterized as the unique maximal partial seminormalization factoring relative normalizations, with the following universal property: any finite modification by a universal homeomorphism that preserves residue fields factors uniquely through $A$8.

Analogs in the real algebraic setting define seminormalization relative to the central locus—the integral closure of the coordinate ring in the ring of hereditarily rational functions on the central locus. This yields a finite birational morphism $A$9 factoring all centrally subintegral extensions (Fichou et al., 2017).

4. Seminormal Bases and Gram Determinants in Representation Theory

In the semisimple case, the seminormal basis for an algebra such as a Hecke algebra, partition algebra, or Temperley–Lieb algebra is constructed via explicit interpolation formulas in Jucys–Murphy elements, yielding a diagonal action. For example, the non-degenerate cyclotomic Hecke algebra of type $a \in A$0 admits a seminormal basis given by $a \in A$1 (with $a \in A$2 the Gelfand–Tsetlin idempotent), and such bases satisfy triangular change-of-basis relations with Murphy (cellular) bases (Hu et al., 11 Dec 2025, Hu et al., 2013).

The Gram determinant for Specht or cell modules can be computed explicitly as products over combinatorial data attached to tableaux—such as the content or residue sequences—yielding cyclotomic polynomial factorizations parameterizing irreducible representations in varying characteristics. This is essential in modular representation theory, in particular for determining decomposition numbers and lifting idempotents to integral forms (Hu et al., 2013, Ryom-Hansen, 2011).

In the context of canonical cellular, graded, and quiver Hecke algebras, the seminormal form underpins the entire homological and categorical structure, for example, by providing explicit bases for projectives and explicit Z-gradings (Ryom-Hansen, 2013).

5. Seminormality and Its Relation to Normality and Weak Normality

Seminormality sits strictly between reducedness and normality; a normal scheme is always seminormal, but the converse is generally false. Weak normality is intermediary: any normal (or topologically normal) space is weakly normal, and any weakly normal scheme is seminormal. The chain of implications is therefore:

$a \in A$3

and similarly for topological normality. These relations are compatible with both scheme-theoretic constructions and module-theoretic representation frameworks (Kollár, 2015).

In real algebraic geometry, seminormalization relative to the central locus arises as a refinement required to manage the topology of real points; over curves, centrally seminormal varieties are precisely those whose complexifications exhibit only ordinary real $a \in A$4-fold singularities, with all tangents and branches real (Fichou et al., 2017).

6. Applications and Examples

Seminormal forms are crucial in both geometry and representation theory:

• Algebraic geometry: Seminormalization controls the failure of descent for algebraic functions and differentials, with $a \in A$5. Cohomological cdh-descent for $a \in A$6 on a reduced scheme $a \in A$7 recovers the sheaf of Kähler differentials on the seminormalization, providing a birational invariant suited for singularities in positive characteristic (Huber et al., 2017).
• Representation theory: Seminormal bases produce explicit primitive idempotents, determine projective covers (e.g., via the LLT-algorithm in modular symmetric group theory), and enable block decompositions even in modular or non-semisimple contexts (Ryom-Hansen, 2013, Bastías et al., 2023, Enyang, 2011). For example, in the Temperley–Lieb algebra, the seminormal idempotents yield a diagrammatic interpretation in terms of Jones–Wenzl projectors and underpin modular structures through $a \in A$8-Jones–Wenzl idempotents (Bastías et al., 2023).

The seminormal approach provides natural bases suited to computational and structural analysis, especially when paired with the cellular or Murphy basis framework compatible with the modern "categorified" perspective on diagram algebras and their categorifications. In the superalgebraic context of Hecke–Clifford algebras, seminormal forms organize both the Clifford and Hecke generator actions, yielding explicit, eigenbasis-level control of simple modules (Li et al., 21 Feb 2025).

7. Cohomological and Birational Implications

The seminormalization construction is intimately connected to the behavior of sheaves under birational and cohomological operations. In positive characteristic, the passage to seminormality aligns with cdh-topological sheafification: not only is $a \in A$9, but

$a^2 = b$0

and similar statements hold for other variants (rh, eh, sdh topologies) (Huber et al., 2017). These equivalences make the seminormalization a natural setting for birational invariants, the study of $a^2 = b$1-singularities, and analytic comparison with Berkovich spaces, where rank-one valuations correspond to points on $a^2 = b$2, and such valuation-theoretic descriptions align with the algebraic seminormalization locus.

In summary, seminormal forms and seminormalization provide a framework for extracting canonical, optimal structures from both algebraic and representation-theoretic objects, mediating between singular and regular settings, and facilitating explicit computational and homological analyses.