--- title: Hemmer–Nakano Dimension in Representation Theory url: https://www.emergentmind.com/topics/hemmer-nakano-dimension type: topic --- # Hemmer–Nakano Dimension in Representation Theory The **Hemmer–Nakano dimension** is a homological invariant in representation theory attached to a Schur functor from a split quasi-hereditary or highest-weight-type cover to the module category of another algebra. It measures the largest degree up to which the Schur functor preserves Ext-groups on a distinguished resolving subcategory, typically the standard-filtered subcategory \(\mathcal F(\Delta)\). In this sense it is a numerical measure of the quality of the connection between the covering algebra and the algebra being resolved, and recent work computes it explicitly for Schur algebras and non-faithful Schur–Weyl duality situations [2208.00291] [2507.04460]. ## 1. Definition and categorical framework The basic setup starts with a commutative Noetherian ring \(R\), a projective Noetherian \(R\)-algebra \(A\), and a finitely generated projective \(A\)-module \(P\). One sets \[ B=\operatorname{End}_A(P)^{\mathrm{op}}, \qquad F=\operatorname{Hom}_A(P,-)\colon A\text{-mod}\to B\text{-mod}. \] The exact functor \(F\) is the Schur functor. In the highest-weight setting, \(A\) is a split quasi-hereditary \(R\)-algebra with standard modules \(\{\Delta(\lambda)\}_{\lambda\in\Lambda}\), and the standard-filtered subcategory is \[ \mathcal{F}(\Delta)=\{M\in A\text{-mod}:\ M \text{ has a finite filtration with subquotients } \Delta(\lambda)\otimes_R U\}. \] The Hemmer–Nakano dimension is then defined for a resolving subcategory \(\mathcal A\subseteq A\text{-mod}\cap R\text{-proj}\) as \[ HNdim_F(\mathcal{A}) = \sup\Bigl\{n\in \mathbb{N}_0:\ \operatorname{Ext}_A^j(M,N)\xrightarrow{\sim}\operatorname{Ext}_B^j(FM,FN)\ \forall\,M,N\in \mathcal{A},\ 0\le j\le n\Bigr\}. \] If \((A,P)\) is not even a \((-1)\)-\(\mathcal A\)-cover, the convention is \[ HNdim_F(\mathcal A)=-\infty. \] For split quasi-hereditary covers, taking \(\mathcal A=\mathcal F(\Delta)\), this is the degree of faithfulness of standard modules in Rouquier’s sense [2208.00291]. | Object | Definition | Role | |---|---|---| | \(A\) | projective Noetherian \(R\)-algebra | covering algebra | | \(P\) | finitely generated projective \(A\)-module | defines the cover | | \(B\) | \(\operatorname{End}_A(P)^{\mathrm{op}}\) | target algebra | | \(F\) | \(\operatorname{Hom}_A(P,-)\) | Schur functor | | \(\mathcal F(\Delta)\) | standard-filtered subcategory | main test class | | \(HNdim_F(\mathcal A)\) | maximal Ext-degree preserved by \(F\) | homological quality measure | A cover \((A,P)\) of \(B\) means that \(F\) is fully faithful on \(A\)-projectives, equivalently that the canonical map \[ A \longrightarrow \operatorname{End}_B(FA)^{\mathrm{op}} \] is an isomorphism. The Hemmer–Nakano dimension refines this double-centralizer property by asking how far Ext-comparison persists beyond degree \(0\) [2208.00291]. ## 2. Faithfulness levels and Ext-comparison The language of \(n\)-faithfulness organizes the invariant. For modules \(M,N\) with a standard filtration, \((A,P)\) is \(n\)-faithful if \[ \operatorname{Ext}_A^i(M,N)\xrightarrow{\sim} \operatorname{Ext}_B^i(FM,FN) \quad\text{for all }0\le i\le n. \] This yields a hierarchy. A \((-1)\)-faithful cover means that the Schur functor is faithful on the chosen class. A \(0\)-faithful cover means that \(F\) is full and faithful on \(\mathcal F(\Delta)\). A \(1\)-faithful cover means that \(F\) induces an exact equivalence \[ \mathcal{F}(\Delta)\xrightarrow{\sim}\mathcal{F}(F\Delta) \] with inverse given by the right adjoint \(G\) restricted appropriately. Higher values record preservation of increasingly long Ext-patterns [2208.00291]. The right adjoint is \[ G=\operatorname{Hom}_B(FA,-). \] In the split quasi-hereditary case the unit \[ \eta_M\colon M\to GFM \] detects low-degree faithfulness: \((A,P)\) is \((-1)\)-faithful iff \(\eta_M\) is monic for all \(M\in \mathcal F(\Delta)\), and \(0\)-faithful iff \(\eta_M\) is an isomorphism for all such \(M\). For \(i\ge 1\), \((A,P)\) is \(i\)-faithful iff \[ R^jG(FM)=0 \quad\text{for all }M\in \mathcal F(\Delta),\ 1\le j\le i. \] This formulation turns Hemmer–Nakano dimension into a vanishing problem for derived adjoints [2208.00291]. A related formulation appears for quasi-hereditary covers arising from Schur–Weyl duality. For a quasi-hereditary algebra \(R_k(n,d)\) with Schur functor \[ F_{n,d}\colon R_k(n,d)\text{-mod}\to \Lambda_k(n,d)\text{-mod}, \] the Hemmer–Nakano dimension of \(\mathcal F(_{R_k(n,d)})\) is the maximal value \(i\in \mathbb N\cup\{\infty\}\) such that for all \(M,N\in \mathcal F(_{R_k(n,d)})\), \[ \operatorname{Ext}_{R_k(n,d)}^j(M,N)\cong \operatorname{Ext}_{\Lambda_k(n,d)}^j(F_{n,d}M,F_{n,d}N) \quad\text{for every }0\le j\le i. \] In this form, the invariant measures the range of cohomological degrees for which the Schur functor induces Ext-isomorphisms between the quasi-hereditary cover and the quotient algebra [2507.04460]. ## 3. Structural bounds and computational principles A central structural theorem gives an upper bound by the size of the simple spectrum of \(B\). If \(R\) is a field and \((A,\{\Delta(\lambda)\}_{\lambda\in\Lambda})\) is split quasi-hereditary, with \[ \Lambda^*=\{\lambda\in\Lambda : FS(\lambda)\neq 0\}, \] then sufficiently high faithfulness forces \(F\) to become an equivalence. More precisely, if \((A,P)\) is a split quasi-hereditary \((d(\Lambda^*)+1)\)-faithful cover of \(B\), then \[ F|_{A\text{-proj}}\colon A\text{-proj}\xrightarrow{\sim} B\text{-proj} \] is an equivalence, and in fact \(F\) is an equivalence of categories. Since \(d(\Lambda^*)+1\le |\Lambda^*|\), this implies \[ HNdim_F(\mathcal{F}(\Delta))\le \#\{\text{simple }B\text{-modules}\}. \] Thus the invariant cannot grow beyond the point at which the cover collapses to Morita-type equivalence [2208.00291]. The main computational mechanism is **relative dominant dimension**. For suitable resolving subcategories \(\mathcal R_A\), the paper proves the inequalities \[ HNdim_F(\mathcal{R}_A)\ge \inf\{\operatorname{domdim}_{(A,R)}M : M\in \mathcal{R}_A\}-2, \] and \[ HNdim_F(\mathcal{R}_A) \le \inf\{\operatorname{domdim}_{(A,R)}M : M\in \mathcal{R}_A\}-2+\dim R. \] This places Hemmer–Nakano dimension between two dominant-dimension expressions, with the ambient ring dimension \(\dim R\) supplying the correction term in the integral setting [2208.00291]. The invariant is also local on the ground ring. If \((A,P)\) is an \(i\)-\(\mathcal R_A\)-cover, then \((S\otimes_R A,S\otimes_R P)\) is the corresponding cover over any flat Noetherian \(R\)-algebra \(S\). Conversely, for regular rings, cover properties can be recovered from localizations and residue fields. A particularly useful criterion states that for a regular local \(R\) with quotient field \(K\), if 1. \((K\otimes_R A,K\otimes_R P)\) is \((i+1)\)-faithful, and 2. for every height-one prime \(\mathfrak p\), the reduction modulo \(\mathfrak p\) is \(i\)-faithful, then \((A,P)\) is already \((i+1)\)-faithful. Dually, quotienting by a projective ideal lowers faithfulness by at most one, and reduction modulo a prime \(\mathfrak p\) of height \(h\) lowers it by at most \(h\) [2208.00291]. ## 4. Schur algebras and integral formulas For \(n\ge d\), the Schur algebra \[ S_R(n,d)=\operatorname{End}_{RS_d}(V_R^{\otimes d}) \] is split quasi-hereditary, with standard modules given by Weyl modules. The Schur functor \[ F=\operatorname{Hom}_{S_R(n,d)}(V^{\otimes d},-) \] links \(S_R(n,d)\)-modules to \(RS_d\)-modules and sends Weyl modules to dual Specht modules. In this setting the Hemmer–Nakano dimension can be computed from dominant-dimension formulas [2208.00291]. The relevant dominant-dimension identities are \[ \operatorname{domdim}(S_R(n,d),R) = \inf\{\,2k\in\mathbb N \mid (k+1)1_R\notin R^\times,\ k0\). The paper also derives \[ \operatorname{Ext}_{\Lambda_k(p,2p)}^l(V^{\otimes 2p},V^{\otimes 2p})=0 \quad\text{for }1\le l\le 4(p-1)-2. \] These vanishings feed into the quasi-hereditary cover theorem, but the paper distinguishes this larger Ext-vanishing range from the final Hemmer–Nakano dimension of the cover itself [2507.04460]. The main numerical result is that for \(p>2\), \[ \boxed{HNdim\bigl(\mathcal F(_{R_k(p,2p)})\bigr)=2(p-2).} \] Equivalently, the Schur functor induces isomorphisms \[ \operatorname{Ext}_{R_k(n,d)}^i(M,N)\cong \operatorname{Ext}_{\Lambda_k(p,2p)}^i(F_{p,2p}M,F_{p,2p}N) \] for all \(M,N\in \mathcal F(_{R_k(n,d)})\) and all \[ 0\le i\le 2(p-2), \] while the isomorphism fails in general for \(i=2(p-2)+1\). The formula \(2(p-2)\) is verified separately for \(p=3\), while \(p=2\) is excluded from the new computation and had already been handled in earlier work [2507.04460]. This calculation is embedded in a broader higher Auslander-theoretic picture. The paper proves that \[ (S_k(p,2p), (k^p)^{\otimes 2p}) \] forms a relative \(4(p-1)\)-Auslander pair, and more precisely that in the principal block \[ (A_0,Q)\ \text{is a relative }4(p-1)\text{-Auslander pair}, \qquad pdim_{A_0} Q=p-2. \] On the \(\Lambda\)-side, \(Q_\Lambda\) is a \((4(p-1),p-2,p-2)\)-quasi-precluster tilting module. Consequences include that the Young module \(Y^p\) has infinite projective dimension, every other non-projective Young module in the principal block has finite projective dimension \(p-2\), and \[ Q_\Lambda/Y^p=\bigoplus_{\lambda\ne p} Y^\lambda \] is a tilting \(\Lambda_0\)-module [2507.04460]. ## 6. Terminological scope and distinction from geometric “Nakano” theories The term **Hemmer–Nakano dimension** belongs to representation theory, Schur functors, quasi-hereditary covers, and Ext-comparison. It should be distinguished from the much older and much broader uses of **Nakano** in complex geometry and vector-lattice theory. Several recent arXiv papers with “Nakano” in the title do not define or use any notion called Hemmer–Nakano dimension. Instead, they concern Nakano curvature positivity, Nakano vanishing, Demailly–Nadel-type vanishing for holomorphic vector bundles, augmented-base-locus refinements of Nakano vanishing, Monge–Ampère-type equations for Nakano positive curvature tensors, Nakano semipositivity of direct image bundles, or Nakano carrier theorems in Riesz spaces [1212.4417] [2011.13653] [2501.05352] [2108.13715] [2503.01666]. This terminological separation matters because the shared word “Nakano” does not signal a shared invariant. In the geometric papers, Nakano refers to positivity or vanishing in complex differential or algebraic geometry, such as the curvature condition \[ i\Theta_h >_{\mathrm{Nak}} \epsilon\,\omega\otimes I \] or variants of Nakano semipositivity and Nakano carrier theory. In the representation-theoretic papers, by contrast, the Hemmer–Nakano dimension is an Ext-theoretic degree bound for a Schur functor. A plausible implication is that the name records historical influence rather than a direct conceptual overlap between these literatures [1212.4417] [2208.00291]. Source: https://www.emergentmind.com/topics/hemmer-nakano-dimension