Papers

This work concerns ideals invariant under the action of the group of linear base changes on a generic matrix; we call these GL-invariant ideals. We determine the ordinary powers, their saturations with respect to determinantal ideals, and the symbolic powers of GL-invariant ideals. We give explicit formulas for asymptotic invariants known as (skew) Waldschmidt constants and asymptotic resurgence, which measure the growth of these families and compare the ordinary and symbolic topologies. We also prove that the generalized symbolic Rees algebras associated with these ideals are Noetherian.

Let \(G\) be a simple and simply connected algebraic group of simply-laced type. For each positive braid word \(\beta\), and for the complete strong duality datum attached to a \(Q\)-datum, we construct an explicit based monoidal categorification of the quantum cluster algebra of the braid variety \(X(\beta)\). We identify this algebra with both a localized quantum Grothendieck ring and a localized level-\(\geq1\) subalgebra of the bosonic extension algebra. Under these identifications, quantum cluster monomials correspond simultaneously to real simple modules and normalized global basis elements. We construct the specialization homomorphism at \(q^{1/2}=1\) and prove that it recovers \(\CC[X(\beta)]\); in particular, the resulting integral form is a flat quantum deformation. We also give intrinsic Lusztig parameters for cluster variables attached to double strings, identify the corresponding quantum grid minors, and establish generalized quantum \(T\)-systems.

The first paper worked out one family of four lattices in full – $|\det| = 12, 24, 36$ and $72$. This paper generalizes it. We enumerate the 44 symmetrizable rank-three hyperbolic Cartan matrices and their 56 depth-one edits, 100 root bases in all, and compute the obstruction space $S_{5/2}(\rho_L)$ of the 98 within our weight-$5/2$ budget: 10 vacuous, 34 unobstructed, 54 obstructed. The vacuous ten are the lattices Bruinier, Ehlen and Freitag call simple, read in signature $(2,3)$. Of their fifteen, our ten realize five; of the rest, five need more than three generators and cannot be the discriminant form of a rank-three lattice at all, three fail $|\det L| = 2k^2$, and two are simply not root bases. Within the rank-three hyperbolic world the simple lattices are exactly the Feingold-Frenkel neighbours of index $k \le 4$. The discriminant group $L'/L$ carries a finite quadratic form, and the finite group of its isometries acts on the weight-$3/2$ cusp forms for $\rho_L$, the bottom antisymmetric rung. We survey the lattices on which that action is absolutely irreducible of dimension at least two, so that the Petersson pairing there is pinned down up to a single scalar. The condition alone cuts the census to three: $L_4$ of the first paper, and two new ones at $|\det| = 40$ and $88$. The quaternion discriminants that occur are 6, 10 and 22, the three for which the Shimura curve $X^D$ has genus zero. Rigidity uses no quaternion input, so we record it as an observation, not a characterisation. Both invariants of the first paper – the shadow norm $\|\Xi\|^2$ and the Petersson scalar $t$ – were single points there. Three rigid lattices instead of one are where their special faces come off: $\|\Xi\|^2$ generalizes against $L(f,1)$ where the first paper read $L(f,2)$, and $t$ against an elliptic curve's imaginary period where it read $\Gamma(1/3)$. Both numerically, to 26-58 digits.

We extend the setup of Lai-Nakano-Xiang's quantum wreath product algebras to symmetric monoidal categories and deformed braid relations. Under some mild assumptions, we drastically simplify the criterion for such algebras to have PBW basis; curiously, the new criterion is phrased in terms of associative and quantum Yang-Baxter equations. As an application, we clean up some basis results in the existing literature, and propose a certain generalization of quiver Hecke algebras.

In this paper, we establish a finite–tame–wild trichotomy for the representation type of $(n+1)$-preprojective algebras of type $A_m$. We also provide a complete classification of such algebras with respect to $\tau$-tilting finiteness. Moreover, for $n\ge 2$, we give a precise characterization of when each indecomposable module is $\tau$-rigid.

Let $G$ be a connected, simply connected complex simple algebraic group and $\mathscr{A}_G=G/U^+$ its base affine space, whose elements are called decorated flags. We introduce the quantum configuration space of decorated flags $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ and initiate its algebraic study, based on the representation theory of quantized enveloping algebras. Our algebra $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ gives a quantum analogue of the configuration space $\mathrm{Conf}_K \mathscr{A}_G$ of $K$ decorated flags, which provides local building blocks for the Fock–Goncharov moduli space $\mathscr{A}_{G,\Sigma}$ of decorated twisted $G$-local systems on a marked surface $\Sigma$. We establish basic algebraic properties of $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ such as the domain property, quantum normalization of representatives, the quantum cyclic shifts, the quantum Wilson lines, whose classical counterparts have been fundamental in the study of $\mathscr{A}_{G,\Sigma}$. Moreover, we construct quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ by transporting the Berenstein–Zelevinsky quantum cluster structure on $\mathcal{O}_q(G)$ via quantum Wilson lines, and prove that $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ coincides with the corresponding quantum cluster algebra and its upper counterpart after the localization at frozen variables. The exchange matrices for our quantum seeds agree with the Goncharov–Shen exchange matrices. We also show that quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ restrict to those for $\mathcal{O}_q(\mathrm{Conf}_3 \mathscr{A}_G)$. Finally, we construct quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$, $K\geq 5$, and prove that the corresponding quantum cluster algebras contain the quantum configuration space $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$.

This paper studies the category $\mathcal{C}$ of grading-restricted $C_1$-cofinite generalized modules for the vertex operator algebra associated to the $W$-algebra $W(2,2)$. We first show that $\mathcal{C}$ is the same as the category of finite length modules whose simple composition factors are the irreducible highest weight $W(2,2)$–modules $L[r]$ of highest weight $(\frac{1-r}{2}, 0)$ for $r \in \mathbb{Z}_{> 0}$, and hence $\mathcal{C}$ carries a braided tensor category structure. Then we prove the fusion rules for the simple objects $L[r]$ are governed by the $\mathfrak{sl}_2$ Clebsch–Gordan rule. In particular, we prove \[ L[r]\boxtimes L[s]\cong \bigoplus_{i=0}^{\min\{r,s\}-1} L[r+s-1-2i]. \] Using the fusion rules and a recent result of Etingof–Penneys, we establish the rigidity of $\mathcal{C}$. We also show the semisimple subcategory generated by the simple objects $L[r]$ is tensor equivalent to the category Rep $\mathfrak{sl}_2$ of finite dimensional $\mathfrak{sl}_2$-modules.

We establish a Serre-type presentation for $\imath$quantum enveloping superalgebras arising from quantum supersymmetric pairs of type A. We first prove, in arbitrary type, that evaluating a quantum Serre polynomial on the coideal generators produces a correction of strictly smaller weight in the natural filtration using the projection technique. We then compute these remainder terms for all type A super Satake diagrams. In particular, we determine the new degree-four relations associated with isotropic odd simple roots of the relevant local Satake diagrams.

We introduce contact symbol systems, a noncommutative analogue of symbol systems for projective fundamental forms, by replacing the polynomial algebra on a vector space by the graded dual of the universal enveloping algebra of a Heisenberg algebra. For a complex projective submanifold equipped with a contact structure, we define contact fundamental forms and prove that, at a general point, they form a contact symbol system, which gives a contact version of the classical result due to E. Cartan. Conversely, we prove that every contact symbol system can be realized as the contact fundamental forms of a projective variety with a dense open Heisenberg orbit, called the Heisenberg-symmetric variety associated to the contact symbol system. We show that the closure of a projectivized nilpotent orbit in a simple Lie algebra is Heisenberg-symmetric if and only if it is the adjoint variety, namely, the projectivization of the minimal nilpotent orbit. For adjoint varieties of non-symplectic simple Lie algebras, we prove the contact analogue of the Landsberg–Manivel strict prolongation property by using Yamaguchi's prolongation theory.

Let $M$ be a silting object in an idempotent complete algebraic triangulated category $\mathcal T$. Put $B={\rm End}_{\mathcal T}(M)$, and let $\mathbb{P}_M\colon {\rm pr}(M)\to K^{[-1,0]}({\rm proj}B)$ be the presentation functor associated with $M$. It was recently asked whether $\mathbb{P}_M(T)$ must be tilting whenever $T\in{\rm pr}(M)$ is a tilting object. We answer this question in the negative by giving an explicit finite-dimensional example. Namely, for $$ \Lambda=k(1\xrightarrow{\alpha}2\xrightarrow{\beta}3\xrightarrow{\gamma}4)/(\alpha\beta\gamma),$$ we construct a silting object $M\in K^b({\rm proj}\Lambda)$ and a tilting object $T=\Sigma\Lambda\in{\rm pr}(M)$ for which $$ {\rm dim}_k{\rm Hom}_{K^b({\rm proj}B)}\bigl(\mathbb{P}_M(T),\Sigma^{-1}\mathbb{P}_M(T)\bigr)=1.$$ Thus $\mathbb{P}_M(T)$ is a two-term silting complex but not a tilting complex.

We determine the measures (in the sense of Harman–Snowden) on the Fraïssé class of partially ordered sets: the space of measures is a union of a plane, eight lines, and 15 isolated points. This is the first case where the space is not equidimensional, and the first primitive case in which it has dimension at least two. ChatGPT was used to obtain many arguments. The writing was done entirely by the author.

It is well known that the Young lattice is the Bratelli diagram of the symmetric groups, expressing how irreducible representations restrict from $\mathfrak{S}_{N}$ to $\mathfrak{S}_{N-1}$. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was identified as the Bratelli diagram of a family of algebras $\{\mathbf{O}_N(X,Y)\}_{N \geq 0}$ by Okada in 1994. In this paper, we first realize the Okada algebra $\mathbf{O}_N(X,Y)$ and the associated monoid $\mathbf{O}_N$ using a labelled version of non-crossing arc-diagrams appearing in the description of the Temperley-Lieb algebra and Jones monoid. We establish, for general parameters $(X,Y)$, that the dimension of the Okada algebra $\mathbf{O}_N(X,Y)$ is $N!$, noting that Okada proved this result only in the semisimple case. We interpret a natural bijection between permutations and labelled arc-diagrams as an incarnation of Fomin's version of the Robinson-Schensted correspondence associated to the Young-Fibonacci lattice. The arc-diagram formalism allow us to probe the structure of the Okada monoid and algebra. In particular we prove that the Okada monoid is a regular, aperiodic $*$-monoid and we describe its Green relations and order. These results allow us to construct a cellular basis of the Okada algebra and to show that $\{\mathbf{O}_N(X,Y)\}_{N \geq 0}$ forms a coherent tower of cellular algebras in the sense of Goodman and Graber. We present some conjectures expressing the Gram determinant of the invariant bilinear form attached to each cell module in terms of Okada's clone Schur functions. We conclude the paper by presenting two follow-up, ongoing projects along with a series of questions pushing further the analogy between the symmetric groups and the Okada algebras.

We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform model studied by Fulman in his thesis. The sharp sparsity threshold for the full rational canonical form coincides with the threshold previously obtained by Lee for finite-field cokernels and by Jung-Lee-Yu for random matrix models over $\mathbb{Z}_p$. Our proof is based on the surjection moment method over the function field $\mathbb{F}_p[t]$, applied to the finite module $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$, whose primary decomposition records the rational canonical form. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold $1/d$ for statistics associated with irreducible polynomials of minimal degree $d$.

We investigate the cohomology of line bundles on the incidence correspondence, the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it. In characteristic zero, this cohomology is governed by the Borel-Weil-Bott theorem. In characteristic p>0, however, it becomes considerably subtler, and admits an equivalent reformulation in terms of cohomology tables for divided powers of the cotangent bundle on projective space. Our approach to the problem involves passing to infinitesimal thickenings of the incidence correspondence inside the ambient product of projective spaces. This leads to recursive formulas for the cohomology, generalizing earlier work of Donkin, of Liu, and of Gao-Raicu. We obtain generating functions for cohomology characters, expressed using truncated Schur polynomials and symmetric polynomials encoding the higher structure constants of the Verlinde algebras of SU(2) at levels (p-2) and (2p-2). Along the way, we exploit two important connections with multiplication in the graded Green-Han-Monsky representation ring: one relates this ring to cohomology, and another connects to the Verlinde algebras through the work of Coulembier-Etingof-Ostrik.

We establish a regularized spectral decomposition of the squared magnitude of a maximal degenerate Eisenstein series as a Schwartz distribution on $\mathrm{GL}_n$. Although the original problem is on $\mathrm{GL}_n$, its spectral components are described entirely by the automorphic spectrum of $\mathrm{GL}_2$. As an application, we prove a partial reciprocity formula relating the second moment of $\mathrm{GL}_n\times\mathrm{GL}_n$ Rankin–Selberg central $L$-values and a mixed moment of $L$-values of $\mathrm{GL}_2$. We also prove, assuming the generalized Ramanujan conjecture, an asymptotic formula for the second moment of the above Rankin–Selberg $L$-functions over a conductor-aspect family with an error term of square-root-cancellation strength.

We define the totally positive part of the coordinate algebra associated to a Cartan datum of simply laced type. We show its relation to the canonical basis of the corresponding plus part of the enveloping algebra.

In this paper, we introduce the controlling $L_{\infty}[1]$-algebra of extended Rota-Baxter algebras. As applications, we obtain the cohomology of extended Rota-Baxter algebras and homotopy extended Rota-Baxter algebras. We also study abelian extensions and formal deformations by the lower cohomology group. Lastly, we describe the intrinsic relationship among extended Rota-Baxter algebras, extended Rota-Baxter Lie algebras, and the eigenvalues of the extended Rota-Baxter operator. As the generalization of Rota-Baxter algebras with weight and modified Rota-Baxter algebras, the corresponding results also hold for those algebras.

This paper deals with polynomiality of algebras of symmetric invariants (or generated by symmetric semi-invariants) associated with some particular Inönü-Wigner contractions in type A. More precisely we are interested with contractions of some parabolic subalgebras in type A with respect to their decomposition into their Levi factor and their nilpotent radical, the latter becoming an abelian ideal of the contraction. When the parabolic subalgebra $\mathfrak p$ has its Levi factor decomposing into three blocks, which are symmetric with respect to the antidiagonal, we show in this article that the algebra of symmetric invariants associated with the contraction of the canonical truncation of $\mathfrak p$ has a Weistrass section. It follows that such an algebra of invariants is a polynomial algebra, for which we can give the number of algebraically independent homogeneous generators, their weight and degree. The method relies on the construction of an adapted pair for such a contraction. When the Levi factor of $\mathfrak p$ has its central block not of the same size as its two extremal blocks, we also show that the above algebra of symmetric invariants is equal to the algebra generated by symmetric semi-invariants associated with the Inönü-Wigner contraction of $\mathfrak p$.

We outline the contours of an emerging theory of Kähler metrics in derived noncommutative geometry. This is a refinement of the theory of Bridgeland stability conditions encoding underlying differential-geometric structures. We propose elements of such a structure in both Archimedean and non-Archimedean settings, including metrized objects, mass measures satisfying a BPS inequality, harmonic metrics, minimizing flows, and complexified Kähler potentials. We develop the framework through examples and constructions involving Fukaya categories, quiver representations and associated C$^*$-algebras, spectral networks, and comonadic adjunctions of stable $\infty$-categories.

We prove the shuffle realization for quantum affine superalgebras of types $B,C,D$ in the loop realization, in that we construct an isomorphism between each of these algebras and a suitably defined double shuffle algebra. We explicitly describe the latter shuffle algebra using certain vanishing conditions that generalize the Feigin-Odesskii wheel conditions; a novel feature is the appearance of a so-called scary wheel, which is a particular order $2$ vanishing condition involving $7$ variables. Along the way, we fully develop the theory of shrubs in super types $A$ (affine and toroidal), which are important combinatorial tools in the study of the shuffle algebras associated to toric Calabi-Yau threefolds. Our techniques also allow us to define quantum toroidal superalgebras of types $B,C,D$. More importantly, we provide a general framework to formulate and prove shuffle realizations in the wide generality of quivers with parameters.

Chen, Liu, Lu, and Zhang recently constructed a pretriangulated category with invertible suspension in which Verdier's octahedral axiom fails. We introduce a general enlargement construction for right pretriangulated categories and show that it preserves axioms (RTR1)-(RTR3), while failure of (RTR4) is detected by the forgetful functor. Applied to their type $A_5$ example, the construction yields a right pretriangulated category that is not right triangulated. In this example, the suspension is faithful but not essentially surjective.

For a $p$-adic field of any residual characteristic, we determine irreducible decompositions of the Weil representation restricted to the maximal open compact subgroups and the Iwahori subgroup by using the Schrödinger model. In particular, these decompositions show that the restrictions are multiplicity free.

Let $G$ be a complex reductive affine algebraic group with its symmetric tensor category $\mathcal{C}_G$ of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion $\operatorname{Ind}(\mathcal{C}_G)$ of $\mathcal{C}_G$ is isomorphic to the algebra $\mathcal{O}(G/H)$ of regular functions on the homogeneous space $G/H$, for some reductive algebraic subgroup $H$ of $G$. Then we apply this result to the theory of vertex operator algebra extensions. In particular, assuming the strong rationality of the $ \operatorname{A}_5$-orbifold of the vertex operator algebra $V_{\mathcal{L}_2}$ associated with the rank-one root lattice $\mathcal{L}_2:= \sqrt{2}\mathbb{Z}$, we classify all the not necessarily rational simple CFT type preunitary vertex operator algebra extensions of the simple unitary Virasoro vertex operator algebra $L(1,0)$ with central charge $c=1$ satisfying a certain spectrum condition. This result is the vertex operator algebra analogue of a conformal net result by Feng Xu. Every strongly rational preunitary vertex operator algebra extension of $L(1,0)$ satisfies the above spectrum condition because of the congruence subgroup modularity property of its characters. As a consequence, we get a complete classification result for strongly rational $c=1$ vertex operator algebras, up to the strong rationality of $V_{\mathcal{L}_2}^{\operatorname{A}_5}$.

We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins–Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided $\Gamma_0$-artinian rings need not be $\Gamma_0$-noetherian, but using our tools we prove that one-sided gr-hereditary $\Gamma_0$-artinian rings, two-sided $\Gamma_0$-artinian rings, and $d$-finitely generated one-sided $\Gamma_0$-artinian rings are $\Gamma_0$-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.

Generic categories arising in representation stability and equivariant commutative algebra are difficult to analyze in positive characteristic, as these categories rarely admit enough (or even any) finitely generated injectives. Our main observation is that they are nonetheless an increasing union of Serre subcategories, each with finitely many simples and enough finite length injectives. We package this into an axiomatic framework which is widely applicable. In particular, we use this to analyze (1) VI-modules in non-describing characteristic, recovering Nagpal's classification of simple generic VI-modules, and (2) GL-equivariant modules over truncated polynomial rings and exterior algebras in infinitely many variables.

Let $G$ be a finite group, let $N$ be a normal subgroup of $G$, and let $\theta$ be an irreducible $p$-Brauer character of $N$, for a prime $p$. It is well known that the number of irreducible Brauer characters of $G$ lying over $\theta$ equals the number of $p$-regular $\theta$-good conjugacy classes of $G_{\theta}/N$. In this note we express the number of real-valued irreducible Brauer characters of $G$ lying over $\theta$ in terms of the $p$-regular $\theta$-good conjugacy classes of $G_{\theta}/N$. This is a modular analogue of one of the main results in a recent paper (arXiv:2301.01177) by John Murray.

We determine two infinite families of poles on the positive real axis for classical single-variable Witten zeta functions. In type $A_r$, for $r \geq 5$, the point $q_r^A = 2(r-4)/(r^2+r-4)$ is a simple pole except in ranks $12$ and $20$, where it is a double pole. In type $D_r$, for $r \geq 4$, the point $q_r^D = (r-3)/(r(r-1)-1)$ is a simple pole except at $D_8$, where it is double. In root-product normalization, we express the simple residues and the leading coefficients of these three double poles in terms of gamma, trigonometric, and Riemann zeta values. For $r \geq 4$, the functions of types $B_r$ and $C_r$ are holomorphic at $q_r^{BC} = (r-3)/(r^2-1)$ except possibly in ranks $7$ and $11$, where any pole is simple. These pole and holomorphy statements arise from quadratic normal Taylor coefficients. For every fixed higher even normal degree, we also determine exactly when the associated continued $B/C$ chamber sum is nonzero; this auxiliary result does not by itself classify poles of the full Witten function. The proofs combine exhaustive support classifications, explicit integration-by-parts identities, and finite chamber relations. No numerical nonvanishing estimate is used.

We define the integral profiles of all modules and introduce integral coefficient rings ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)$ for all finite-dimensional complex algebras $A$. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules $M\cong N$ are isomorphic if and only if their integral profiles are similar, i.e., $M\cong N$ if and only if $\displaystyle \int M \sim \int N$. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of $A$ (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of $A$ is finite if and only if the embedded integral profile of $\mathrm{top}(A)$ lies in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$; (6) the finitistic dimension of $A$ is finite if and only if each embedded integral profile of $M$ lying in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$ implies that its degree is less than or equal to a fixing integer $d\in\mathbb{N}^+$; (7) we provide two sufficient conditions, such that if a finite-dimensional complex algebra $A$ satisfies one of them, then its finitistic dimension is finite.

Let $(K,\mathcal{O},k)$ be a large enough $2$-modular system. We show that there is a block algebra $B$ of $\mathcal{O}{\rm SU}_3(2)$ which is Morita equivalent to $\mathcal{O} Q_8$ via a bimodule with a $3$-dimensional endotrivial source. We also prove that $B$ and $\mathcal{O} Q_8$ are not globally isotypic. Hence they are not $p$-permutation equivalent and not splendidly Rickard equivalent.

We study heat kernels, sub-Riemannian distances, and spectral zeta functions of sub-Laplacians determined by closed connected subgroups of compact Lie groups. Combining Hall's inversion formula with the affine lattice expansion of the compact group heat kernel, we derive a Cartan integral representation involving the group's exponential lattice. For two-step compact Lie pairs, the full algebraic small-time heat trace expansion is determined, up to an exponentially small remainder, by two explicit constants $C_{G,L}$ and $\beta_{G,L}$. This expansion determines all heat coefficients, the poles and residues of the reduced spectral zeta function, and its values at nonpositive integers. For the transvective symmetric subclass, we prove uniform vertical asymptotics for the Carnot-Carathéodory distance; the leading coefficient $\mathfrak F_{G,K}(Z)$ is the attained minimum of a finite-dimensional singular value problem. For simply connected two-step pairs, we obtain an exact decomposition of the zeta-regularized determinant into local, lattice, and spectral terms, with exponential truncation estimates. We specialize these results to block subgroups of $\mathrm{SU}(N)$, recovering the classical $\mathrm{SU}(2)$ and CR sphere spectra.