A Categorification of $\displaystyle {\mathfrak q} (2)$-Crystals

We provide a categorification of \(\mathfrak {q}(2)\)-crystals on the singular \(\mathfrak {gl}_{n}\)-category \({\mathcal O}_{n}\). Our result extends the \(\mathfrak {gl}_{2}\)-crystal structure on \(\text {Irr} ({\mathcal O}_{n})\) induced from the work of Bernstein-Frenkel-Khovanov. Further properties of the \({\mathfrak q}(2)\)-crystal \(\text {Irr} ({\mathcal O}_{n})\) are also discussed.     

On the admissible sets of type $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ over recursively saturated models

Some effective expression is obtained for the elements of an admissible set \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) as template sets. We prove the Σ-reducibility of \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) to \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) for each recursively saturated model \(\mathfrak{M}\) of a regular theory, give a criterion for uniformization in \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) for each recursively saturated model \(\mathfrak{M}\), and establish uniformization in \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{N})\) and \(\mathbb{H}\mathbb{Y}\mathbb{P}(\Re ')\), where \(\mathfrak{N}\) and \(\Re '\) are recursively saturated models of arithmetic and real closed fields. We also prove the absence of uniformization in \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) and \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) for each countably saturated model \(\mathfrak{M}\) of an uncountably categorical theory, and give an example of this type of theory with definable Skolem functions. Furthermore, some example is given of a model of a regular theory with Σ-definable Skolem functions, but lacking definable Skolem functions in every extension by finitely many constants.     

Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Branching Rules for Finite-Dimensional $\mathcal {U}_{q}(\mathfrak {su}(3))$-Representations with Respect to a Right Coideal Subalgebra

We consider the quantum symmetric pair \((\mathcal {U}_{q}(\mathfrak {su}(3)), \mathcal {B})\) where \(\mathcal {B}\) is a right coideal subalgebra. We prove that all finite-dimensional irreducible representations of \(\mathcal {B}\) are weight representations and are characterised by their highest weight and dimension. We show that the restriction of a finite-dimensional irreducible representation of \(\mathcal {U}_{q}(\mathfrak {su}(3))\) to \(\mathcal {B}\) decomposes multiplicity free into irreducible representations of \(\mathcal {B}\). Furthermore we give explicit expressions for the highest weight vectors in this decomposition in terms of dual q-Krawtchouk polynomials.     

Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains \(\Omega \) obtained by different approaches. In particular, we consider three settings \(\mathcal {M}_{u,p}(\Omega )\), \(\mathbb {M}_{u,p}(\Omega )\) and \(\mathfrak {M}_{u,p}(\Omega )\), where \(0<p\le u<\infty \), commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes \(\mathfrak {E}_{\mathsf {G}}(\mathcal {M}_{u,p}(\Omega ))\) as well as \(\mathfrak {E}_{\mathsf {G}}(\mathfrak {M}_{u,p}(\Omega ))\), and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether \(\frac{n}{u}\ge \frac{1}{p}\) or \(\frac{n}{u} < \frac{1}{p}\) plays a decisive role when it comes to the behaviour of these spaces.     

Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, and \(Y_{\hbar }(\mathfrak{g})\), \(U_{q}(L\mathfrak{g})\) the corresponding Yangian and quantum loop algebra, with deformation parameters related by \(q=e^{\pi \iota \hbar }\). When \(\hbar \) is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor \(\Gamma \) from the category of finite-dimensional representations of \(Y_{\hbar }(\mathfrak{g})\) to those of \(U_{q}(L \mathfrak{g})\). The functor \(\Gamma \) is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of \(\operatorname{Rep}_{\operatorname{fd}}(Y_{\hbar }(\mathfrak{g}))\) defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on \(\Gamma \) and show that, if \(|q|\neq 1\), it yields an equivalence of meromorphic braided tensor categories, when \(Y_{\hbar }(\mathfrak{g})\) and \(U_{q}(L\mathfrak{g})\) are endowed with the deformed Drinfeld coproducts and the commutative part of their universal \(R\)-matrices. This proves in particular the Kohno–Drinfeld theorem for the abelian \(q\)KZ equations defined by \(Y_{\hbar }(\mathfrak{g})\). The tensor structure arises from the abelian \(q\)KZ equations defined by an appropriate regularisation of the commutative part of the \(R\)-matrix of \(Y_{\hbar }(\mathfrak{g})\).     

Generators of the ring of weakly holomorphic modular functions for $$\Gamma _1(N)$$

For a positive integer N divisible by 4, 5, 6, 7 or 9, let \(\mathcal {O}_{1,N}(\mathbb {Q})\) be the ring of weakly holomorphic modular functions for the congruence subgroup \(\Gamma _1(N)\) with rational Fourier coefficients. We present explicit generators of the ring \(\mathcal {O}_{1,N}(\mathbb {Q})\) over \(\mathbb {Q}\) by making use of modular units which have infinite product expansions.     

Positive Systems of Kostant Roots

Let \(\mathfrak {g}\) be a simple complex Lie algebra and let \(\mathfrak {t} \subset \mathfrak {g}\) be a toral subalgebra of \(\mathfrak {g}\). As a \(\mathfrak {t}\)-module \(\mathfrak {g}\) decomposes as

$$\mathfrak{g} = \mathfrak{s} \oplus \left( \oplus_{\nu \in \mathcal{R}}~ \mathfrak{g}^{\nu}\right)$$

where \(\mathfrak {s} \subset \mathfrak {g}\) is the reductive part of a parabolic subalgebra of \(\mathfrak {g}\) and \(\mathcal {R}\) is the Kostant root system associated to \(\mathfrak {t}\). When \(\mathfrak {t}\) is a Cartan subalgebra of \(\mathfrak {g}\) the decomposition above is nothing but the root decomposition of \(\mathfrak {g}\) with respect to \(\mathfrak {t}\); in general the properties of \(\mathcal {R}\) resemble the properties of usual root systems. In this note we study the following problem: “Given a subset \(\mathcal {S} \subset \mathcal {R}\), is there a parabolic subalgebra \(\mathfrak {p}\) of \(\mathfrak {g}\) containing \(\mathcal {M} = \oplus _{\nu \in \mathcal {S}} \mathfrak {g}^{\nu }\) and whose reductive part equals \(\mathfrak {s}\)?”. Our main results is that, for a classical simple Lie algebra \(\mathfrak {g}\) and a saturated \(\mathcal {S} \subset \mathcal {R}\), the condition \((\text {Sym}^{\cdot }(\mathcal {M}))^{\mathfrak {s}} = \mathbb {C}\) is necessary and sufficient for the existence of such a \(\mathfrak {p}\). In contrast, we show that this statement is no longer true for the exceptional Lie algebras F₄,E₆,E₇, and E₈. Finally, we discuss the problem in the case when \(\mathcal {S}\) is not saturated.

     

Irreducible Modules Over Witt Algebras $\mathcal {W}_{n}$ and Over 𝖘𝖑n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$

In this paper, by using the “twisting technique” we obtain a class of new modules A _b over the Witt algebras \(\mathcal {W}_{n}\) from modules A over the Weyl algebras \(\mathcal {K}_{n}\) (of Laurent polynomials) for any \(b\in \mathbb {C}\). We give necessary and sufficient conditions for A _b to be irreducible, and determine necessary and sufficient conditions for two such irreducible \(\mathcal {W}_{n}\)-modules to be isomorphic. Since \(\mathfrak {sl}_{n+1}(\mathbb {C})\) is a subalgebra of \(\mathcal {W}_{n}\), all the above irreducible \(\mathcal {W}_{n}\)-modules A _b can be considered as \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules. For a class of such \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules, denoted by Ω_1?a (λ ₁, λ ₂, ? ,λ _n ) where \(a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}\), we determine necessary and sufficient conditions for these \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules to be irreducible. If the \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module Ω_1?a (λ ₁, λ ₂,? ,λ _n ) is reducible, we prove that it has a unique nontrivial submodule W _1?a (λ ₁, λ ₂,...λ _n ) and the quotient module is the finite dimensional \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module with highest weight mΛ _n for some non-negative integer \(m\in \mathbb {Z}_{+}\). We also determine necessary and sufficient conditions for two \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules of the form Ω_1?a (λ ₁, λ ₂,? ,λ _n ) or of the form W _1?a (λ ₁, λ ₂,...λ _n ) to be isomorphic.     

Zigzag Strip Bundle Realization of B(Λ0) over $U_{q}(C_{n}^{(1)})$

Zigzag strip bundles are new combinatorial models realizing the crystals B(∞) for the quantum affine algebras \(U_{q}(\mathfrak {g})\), where \(\mathfrak {g}=B_{n}^{(1)},D_{n}^{(1)}\), \(D_{n+1}^{(2)}\), \(C_{n}^{(1)}\), \(A_{2n-1}^{(2)}\), \(A_{2n}^{(2)}\). Recently, these models were used to the realization of highest weight crystals except for the highest weight crystal B(Λ₀) over the quantum affine algebra \(U_{q}(C_{n}^{(1)})\). In this paper, we construct the highest weight crystal B(Λ₀) over the quantum affine algebra \(U_{q}(C_{n}^{(1)})\) using zigzag strip bundles, which completes the realizations of all highest weight crystals over \(U_{q}(\mathfrak {g})\).     

Hall Polynomials for Nakayama Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k. In this note we first show that the existence of Hall polynomials for \(\mathcal{A}\) equivalent to the existence of the Hall polynomial \(\varphi^{M}_{N L}\) for each \(M, L \in mod\mathcal{A}\) and \(N\in ind\mathcal{A}\). Then we show that for a basic connected Nakayama algebra \(\mathcal{A}\), \(\mathcal{H}(\mathcal{A})=\mathcal{L}(\mathcal{A})\) and Hall polynomials exist for this algebra. We also provide another proof of the existence of Hall polynomials for the representation directed split algebras.     

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

On counting subring-submodules of free modules over finite commutative frobenius rings

Let \(\texttt {R}\) be a finite commutative Frobenius ring and \(\texttt {S}\) a Galois extension of \(\texttt {R}\) of degree m. For positive integers k and \(k'\), we determine the number of free \(\texttt {S}\)-submodules \(\mathcal {B}\) of \(\texttt {S}^\ell \) with the property \(k=\texttt {rank}_\texttt {S}(\mathcal {B})\) and \(k'=\texttt {rank}_\texttt {R}(\mathcal {B}\cap \texttt {R}^\ell )\). This corrects the wrong result (Bill in Linear Algebr Appl 22:223–233, 1978, Theorem 6) which was given in the language of codes over finite fields.     

On rational isomorphisms of Lie algebras

Let \(\mathfrak{n}\) be a finite-dimensional noncommutative nilpotent Lie algebra for which the ring of polynomial invariants of the coadjoint representation is generated by linear functions. Let \(\mathfrak{g}\) be an arbitrary Lie algebra. We consider semidirect sums \(\mathfrak{n} \dashv _\rho \mathfrak{g}\) with respect to an arbitrary representation ρ: \(\mathfrak{g}\) → der \(\mathfrak{n}\) such that the center z \(\mathfrak{n}\) of \(\mathfrak{n}\) has a ρ-invariant complement.We establish that some localization \(\tilde P(\mathfrak{n} \dashv _\rho \mathfrak{g})\) of the Poisson algebra of polynomials in elements of the Lie algebra \(\mathfrak{n} \dashv _\rho \mathfrak{g}\) is isomorphic to the tensor product of the standard Poisson algebra of a nonzero symplectic space by a localization of the Poisson algebra of the Lie subalgebra \((z\mathfrak{n}) \dashv \mathfrak{g}\). If \([\mathfrak{n},\mathfrak{n}] \subseteq z\mathfrak{n}\), then a similar tensor product decomposition is established for the localized universal enveloping algebra of the Lie algebra \(\mathfrak{n} \dashv _\rho \mathfrak{g}\). For the case in which \(\mathfrak{n}\) is a Heisenberg algebra, we obtain explicit formulas for the embeddings of \(\mathfrak{g}_P \) in \(\tilde P(\mathfrak{n} \dashv _\rho \mathfrak{g})\). These formulas have applications, some related to integrability in mechanics and others to the Gelfand-Kirillov conjecture.     

Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

In this article, we consider the following fractional Hamiltonian systems:

$$\begin{aligned} {_{t}}D_{\infty }^{\alpha }({_{-\infty }}D_{t}^{\alpha }u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb {R}, \end{aligned}$$

where \(\alpha \in (1/2, 1)\), \(\lambda >0\) is a parameter, \(L\in C(\mathbb {R}, \mathbb {R}^{n\times n})\) and \(W \in C^{1}(\mathbb {R} \times \mathbb {R}^n, \mathbb {R})\). Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all \(t\in \mathbb {R}\), that is, \(L(t) \equiv 0\) is allowed to occur in some finite interval \(\mathbb {I}\) of \(\mathbb {R}\). Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on \(\mathbb {R} \setminus \mathbb {I}\) as \(\lambda \rightarrow \infty ,\) and converge to \(\tilde{u}\) in \(H^{\alpha }(\mathbb {R})\); here \(\tilde{u} \in E_{0}^{\alpha }\) is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval \(\mathbb {I}\). Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.

     

On $$\mathfrak {F}_{hq}$$-supplemented subgroups of a finite group

A subgroup H of a finite group G is quasinormal in G if it permutes with every subgroup of G. A subgroup H of a finite group G is \(\mathfrak {F}_{hq}\)-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and \((H\cap N)H_{G}/ H_{G} \le Z_{\mathfrak {F}}(G/H_{G})\), where \(H_{G}\) is the core of H in G and \({Z}_{\mathfrak {F}} (G/H_{G})\) is the \(\mathfrak {F}\)-hypercenter of \({G/H}_{G}\). This paper concerns the structure of a finite group G under the assumption that some subgroups of G are \(\mathfrak {F}_{hq}\)-supplemented in G.     

Some results on the annihilators and attached primes of local cohomology modules

Let \((R, \mathfrak {m})\) be a local ring and M a finitely generated R-module. It is shown that if M is relative Cohen–Macaulay with respect to an ideal \(\mathfrak {a}\) of R, then \({\text {Ann}}_R(H_{\mathfrak {a}}^{{\text {cd}}(\mathfrak {a}, M)}(M))={\text {Ann}}_RM/L={\text {Ann}}_RM\) and \({\text {Ass}}_R (R/{\text {Ann}}_RM)\subseteq \{\mathfrak {p}\in {\text {Ass}}_R M|\,\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})={\text {cd}}(\mathfrak {a}, M)\},\) where L is the largest submodule of M such that \(\mathrm{cd}(\mathfrak {a}, L)< \mathrm{cd}(\mathfrak {a}, M)\). We also show that if \(H^{\dim M}_{\mathfrak {a}}(M)=0\), then \({\text {Att}}_R(H^{\dim M-1}_{\mathfrak {a}}(M))= \{\mathfrak {p}\in {\text {Supp}}(M)|\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\dim M-1\},\) and so the attached primes of \(H^{\dim M-1}_{\mathfrak {a}}(M)\) depend only on \({\text {Supp}}(M)\). Finally, we prove that if M is an arbitrary module (not necessarily finitely generated) over a Noetherian ring R with \(\mathrm{cd}(\mathfrak {a}, M)=\mathrm{cd}(\mathfrak {a}, R/{\text {Ann}}_RM)\), then \({\text {Att}}_R(H^{\mathrm{cd}(\mathfrak {a}, M)}_{\mathfrak {a}}(M))\subseteq \{\mathfrak {p}\in {\text {V}}({\text {Ann}}_RM)|\,\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\mathrm{cd}(\mathfrak {a}, M)\}.\) As a consequence of this, it is shown that if \(\dim M=\dim R\), then \({\text {Att}}_R(H^{\dim M}_{\mathfrak {a}}(M))\subseteq \{\mathfrak {p}\in {\text {Ass}}_R M|\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\dim M\}\).     

Symmetric quiver Hecke algebras and <Emphasis Type="Italic">R</Emphasis>-matrices of quantum affine algebras IV

Let \(U'_q(\mathfrak {g})\) be a twisted affine quantum group of type \(A_{N}^{(2)}\) or \(D_{N}^{(2)}\) and let \(\mathfrak {g}_{0}\) be the finite-dimensional simple Lie algebra of type \(A_{N}\) or \(D_{N}\). For a Dynkin quiver of type \(\mathfrak {g}_{0}\), we define a full subcategory \({\mathcal C}_{Q}^{(2)}\) of the category of finite-dimensional integrable \(U'_q(\mathfrak {g})\)-modules, a twisted version of the category \({\mathcal C}^{(1)}_{Q}\) introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur–Weyl duality, we construct an exact faithful KLR-type duality functor \({\mathcal F}_{Q}^{(2)}:\mathrm{Rep}(R) \rightarrow {\mathcal C}_{Q}^{(2)}\), where \(\mathrm{Rep}(R)\) is the category of finite-dimensional modules over the quiver Hecke algebra R of type \(\mathfrak {g}_{0}\) with nilpotent actions of the generators \(x_k\). We show that \({\mathcal F}_{Q}^{(2)}\) sends any simple object to a simple object and induces a ring isomorphism Open image in new window .     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.