Papers

Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, and let $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$. Let $\mathrm{H}_{1,n-1}=\{\mathrm{diag}(g_1,g_2)\in\mathrm{G}_1,\ g_2\in\mathrm{G}_{n-1}\}$ and, for $s \in \mathbb{R}$, define $\chi_s(\mathrm{diag}(g_1,g_2))=\nu(g_1)^{2s}\nu(g_2)^{-2s}$. We classify, for $n=3,4$, the irreducible smooth representations of $\mathrm{G}_n$ admitting a generalized linear period with respect to $(\mathrm{H}_{1,n-1},\chi_s)$. Motivated by these results, we conjecture a complete classification for all $n>2$. Assuming this conjecture, we characterize such representations in terms of Langlands parameters: an irreducible smooth representation $\pi$ of $\mathrm{G}_n$ admits such a period if and only if $\mathfrak{L}(\pi)$ contains a Weil-Deligne subrepresentation isomorphic to $\mathfrak{L}(\nu^{-2s})$, the $(2n-4)$-dimensional parameter of $\nu^{-2s}$ on $\mathrm{G}_{n-2}$, and the four-dimensional quotient is the Langlands parameter of either the trivial representation of $\mathrm{G}_2$, with $s=\pm\frac{n-2}{2}$, or an irreducible infinite-dimensional $\mathrm{H}_{1,1}$-distinguished representation of $\mathrm{G}_2$. We also verify the Lapid-Prasad conjecture in this setting: the $L$-packet of an irreducible representation of $\mathrm{G}_n$ admitting a linear period with respect to $\mathrm{H}_{1,n-1}$ is invariant under $\rho\mapsto\widetilde{\rho}^{\theta}$.

Canonical ideal quotients of extriangulated categories admit natural one-sided triangulated or pretriangulated structures. A fundamental question is whether these quotient structures still retain enough information to recover higher orthogonality properties of the subcategory being factored out. We show that, for a strongly functorially finite $n$-rigid subcategory, such information is encoded by the nilpotency of the canonical suspension and, equivalently, of the canonical loop. More precisely, this nilpotency characterizes two-sided maximal $n$-orthogonality. We further establish an objectwise recognition criterion that combines one-sided higher orthogonality with the vanishing of the $n$th suspension or loop. In the triangulated setting, our results give a converse to the known truncation construction and show that the quotient is $n$-truncated if and only if the subcategory is $(n+1)$-cluster tilting. Moreover, in the nonsplit case, the nilpotency index is exactly $n$. These results turn truncation from a consequence of higher orthogonality into a sharp criterion for recognizing it.

We introduce the notion of a quasi-cellular category, which is a variant of cellular algebras and cellular categories. Many diagram categories, such as the Brauer category and the partition category, are quasi-cellular. We study the Brauer category $B$ over a commutative ring $\mathbb k$ with parameter $\delta$ through its quasi-cellular structure. We construct a linear functor $L:B\to mB$ to the matrix Brauer category $mB$, which has the same objects and hom-spaces as $B$ but a matrix-like composition. For a non-singular parameter $\delta$, we show that $L$ is an isomorphism of linear categories, and that $B$ is Morita equivalent to its subcategory $\mathbb S$ spanned by permutations, whose endomorphism algebras are the group algebras of symmetric groups. These results generalize those of Brown and König and Xi for the Brauer algebras.

Let $\pi$ be an irreducible generic representation of $\operatorname{GL}_n(\mathbb{C})$. We classify the exceptional poles of the associated Asai $L$-function $L(s,\pi,\mathrm{As})$ and give a representation-theoretic characterization of them. When $\pi$ is in general position, we further express $L(s,\pi,\mathrm{As})$ in terms of the exceptional $L$-factors attached to the irreducible constituents occurring in the derivatives of $\pi$.

We compute the missing three-color relation in type $H_3$ as an explicit $R$-linear combination of double leaves. The expression has $43\,424$ nonzero polynomial coefficients with scalars in $\mathbb{Z}[\varphi]$, where $\varphi = (1 + \sqrt{5})/2$. We verify the relation using exact arithmetic in the computer algebra system GAP. This completes the diagrammatic presentation of the Hecke category of Elias and Williamson.

We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions $W_{\mathcal A}$. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.

We reduce the Alperin-McKay-Navarro conjecture to an inductive condition on the universal covering groups of non-abelian finite simple groups. The reduction yields height-zero character bijections compatible with Brauer correspondence and block relations between H-triples for arbitrary finite ambient groups. We record the resulting arithmetic and structural consequences, including preservation of $p$-rationality levels and character-theoretic criteria for Sylow subgroups. The proof combines semilinear centralization and transfer through quasisimple components with induction on the central index. It uses the Clifford and gluing theorems for block relations and a central-defect Dade-Glauberman-Nagao correspondence established in two related papers. We also verify the inductive condition for several sporadic groups at primes for which the Sylow subgroups have prime order, and in defining characteristic for the Suzuki groups ${}^{2}B_{2}(2^{2m+1})$ with $m \geq 2$ and the small Ree groups ${}^{2}G_{2}(3^{2m+1})$ with $m \geq 1$.

We study block relations between H-triples of finite groups. We prove a Clifford correspondence theorem in which a single pair of associated projective representations satisfies both the mixed comparison-function identities and the block conditions on every intermediate group. The resulting correspondence preserves relative character heights and yields a common defect group for each pair of corresponding blocks. We also establish induction and gluing results, including an induction criterion that allows unequal Clifford indices. Direct products, wreath products and changes of ambient group are treated within the same framework. These results provide character-triple tools for Galois-equivariant local-global correspondences.

We prove that the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ is classically free at the boundary admissible level, which confirms a conjecture of Andrews, van Ekeren, and Heluani.

We classify the Classifying Grothendieck rings of pivotal fusion categories of rank four over the complex field. There are exactly fifteen, each admitting a unitary categorification. Central induction and Frobenius-Schur indicators give a uniform Frobenius-Perron dimension bound of 3600. The finite classification combines new arithmetic and twist obstructions with an exhaustive census and exact exclusion certificates.

Strict unimodality of Ooms spectra fails for Frobenius parabolic Lie algebras beyond the maximal-parabolic class. We construct explicit Frobenius parabolics whose principal multiplicity sequences contain arbitrarily long outward rises: for every $L\ge1$ there are $L$ consecutive negative first differences, each of quadratic size in $L$. In a simple one-parameter subfamily the transition to nonunimodality occurs in $\slie_{136}$, where two adjacent multiplicities are $541<542$. We also give a representation-theoretic criterion for unimodality. If $F$ is a Frobenius functional with semisimple integral principal element $H$ and $\g=\bigoplus_j\g_j$, the Kirillov form makes the centered grading symplectic and its character has the unique expansion \[ \Theta_\g(x)=\sum_{m\ge1}(\dim\g_m-\dim\g_{m+1})\,\chi_{2m-1}(x) \] in irreducible $SL_2$ characters. Thus Ooms unimodality is equivalent to effectivity of this virtual $SL_2$ character, or equivalently to extension of the centered principal cocharacter to a symplectic $SL_2$ action. For Frobenius seaweeds the unbroken-spectrum theorem says that, after centering, every odd weight in the support occurs; strict unimodality asks for the stronger positivity of every irreducible coefficient. A companion paper proves that stronger property for every Frobenius maximal parabolic. The families constructed here show that beyond that class there can be arbitrarily long blocks of negative irreducible coefficients, with unbounded total negative mass.

Borel's theorem presents the flag manifold cohomology as the coinvariant algebra $H^*(K/T; \mathbb{C}) \cong S(\mathfrak{h}^\vee)/I_W$, establishing the vanishing of characteristic forms $p_j(F_\mathfrak{t}) = \mathrm{tr}(F_\mathfrak{t}^j)$ non-constructively; projection from the Weil algebra $W(\mathfrak{k})$ yields closed primitives on $K$ failing to descend. We construct relative transgression primitives $\alpha_j \in (\Lambda^{2j-1}\mathfrak{m}^\vee)^T \cong {}^K\Omega^{2j-1}(K/T)$ satisfying $d_\mathfrak{m} \alpha_j = -p_j(F_\mathfrak{t})$ on $K/T$. Via connections $\theta_s = \theta_\mathfrak{t} + s\theta_\mathfrak{m}$, the curvature factors as $F_s = (1-s)(F_\mathfrak{t} - s\theta_\mathfrak{m}^2)$; expanding via non-commutative word sums and Euler's Beta integral yields a universal closed formula for all $j \ge 1$. These primitives exhibit a structural transition: $\alpha_2 = -\frac{1}{3}\mathrm{tr}(\theta_\mathfrak{m}^3)$ is curvature-free, $\alpha_3$ is linear in $F_\mathfrak{t}$, while at $j = 4$, $[F_\mathfrak{t}, \theta_\mathfrak{m}^2] \neq 0$ shatters cyclic trace symmetry into independent necklaces, producing the irreducible invariant $\mathrm{tr}(\theta_\mathfrak{m} \wedge F_\mathfrak{t} \wedge \theta_\mathfrak{m}^2 \wedge F_\mathfrak{t})$. On the $A_2$ root lattice ($\mathfrak{sl}_3(\mathbb{C})$), $T$-invariant cochains correspond to closed loops, $\alpha_2 = \Omega_- - \Omega_+$ is the oriented difference of triangles, and $d_\mathfrak{m}$ acts as an edge-fracturing operator witnessing the exactness of $c_2(E)$. While character theory encounters a multiplicity barrier at $k=4$ in Specht decompositions, a minimal Koszul DGA resolution $K_\bullet(p_2, \dots, p_k)$ resolves the coinvariant algebra, recovering the Poincaré polynomial $P_{K/T}(t) = \prod_{j=2}^k \frac{1-t^{2j}}{1-t^2}$ and Euler characteristic $\chi(K/T) = k!$ without matrix-rank ambiguities.

We prove universal positivity of rank-two quantum greedy elements when $b\mid c$ or $c\mid b$, resolving a conjecture of Lee, Li,Rupel and Zelevinsky.

Let $X=G/H$ be a real reductive symmetric space. We consider the support of the Plancherel measure for the regular representation of $G$ on $L^2(X)$, and its Fell topology. We first recall how the Plancherel formula for $X$ (known from work of Delorme and van den Ban–Schlicktrull, among others) yields a description of the support in terms of the discrete spectra of smaller symmetric spaces. We then point out that this description simplifies for a class of real symmetric spaces $X$ which we call well-tempered. These are the spaces whose tangent space at the identity coset admits a Cartan subspace whose centralizer in the Lie algebra of $G$ is abelian. Well-tempered spaces are tempered in the sense of Benoist and Kobayashi. Under mild assumptions on the disconnectedness of $G$, we give a description of the support of the Plancherel measure for well-tempered symmetric spaces in terms of characters of Cartan subgroups of $G$, and describe the Fell topology of the support. If $G$ is a complex group and $H$ a real form, then the support decomposes as a union of explicit quotients of relative continuous-parameter spaces by finite Weyl groups. As an application, we describe the $C^*$-algebra associated to the regular representation of $G$ on $L^2(X)$, and its $K$-theory.

Let $B$ be a finite-dimensional algebra, $M\in\modu B$, and $A=B[M]$ the one-point extension. We give a stable two-step extension criterion for the new simple $A$-module to have derived delooping level at most one. For a concrete algebra $B$ with a self-injective Nakayama quotient $R$, we determine the derived and classical delooping levels of $B[M]$ for every nonzero $M\in\modu R$. In particular, we obtain an explicit algebra $A$ satisfying \[ \Findim(A^{\mathrm{op}})=1<2=\ddell A=\dell A, \] which gives a negative answer to a question of Guo and Igusa.

We prove that any block of a profinite group with a topologically finitely generated defect group $D$ is isomorphic to a block of a virtually pro-$p$ group, and if $D$ is finitely presented then so is the basic algebra of the block over $k$. Using these general results we then prove that any block with defect group $D=\mathbb Z_p^n$ is Morita equivalent to $\mathcal O_\alpha[[\mathbb Z_p^n\rtimes E]]$ for a finite $p'$-group $E$ acting faithfully and a $2$-cocycle $\alpha\in H^2(E,k^\times)$. In particular, Donovan's conjecture holds for such blocks of profinite groups.

In this paper, we prove that for $\ell>5$, the number of $\ell$-regular conjugacy classes of a finite group is at least that of the normalizer of a Sylow $\ell$-subgroup.

We construct Laurent phenomenon structures on reductive branching algebras. For $A_{2n-1}\downarrow C_{n}$, $n\ge 2$, and the exceptional inclusions $D_4\downarrow G_2$, $E_6\downarrow F_4$, and $F_4\downarrow B_4$, we identify the branching algebras with upper Laurent phenomenon algebras, keeping the frozen coefficients polynomial. Degree-fibred categories of projective presentations give a common construction. It also gives cluster seeds of type $A_1$ for $B_3 \downarrow G_2$ and $G_2\downarrow A_2$. We give uniform sufficient conditions identifying a specialized Keel–Yu mirror algebra with an upper Laurent phenomenon algebra admitting a suitable binomial seed. The resulting theta basis is simultaneously adapted to the frozen boundary valuations. For $A_{2n-1}\downarrow C_{n}$ with $2\le n\le5$ and for the other exceptional inclusions, we obtain homogeneous theta bases parametrized by lattice points of explicit rational polyhedral cones, whose weight fibres compute every branching multiplicity. We also prove mutation invariance of finite upper bounds for LP patterns over a UFD.

For a reduced crystallographic root system with Weyl group $W$, Weyl alternants are signed sums of exponentials indexed by $W$. They occur in the numerators of the Weyl character formula and the explicit formula for spherical functions on complex semisimple groups. We establish rootwise bounds that quantify cancellation in these oscillatory sums, with each positive root contributing a factor depending on both the spatial variable and the spectral parameter. More generally, we establish rootwise bounds for all mixed directional derivatives of normalized Weyl alternants. Our proof extends to arbitrary rank the descent strategy used by the second author for pointwise character bounds on $\mathrm{SU}(3)$. The key ingredients include the decomposition of the Weyl group into parabolic double cosets and the relative BGG–Demazure identity. For spherical functions on complex semisimple groups, these estimates recover the recent pointwise bound of Brumley, Marshall, Matz, and Peterson, and they refine the derivative bounds of Cowling and Nevo by controlling arbitrary mixed radial derivatives uniformly across spatial and spectral root hyperplanes. Furthermore, by a standard localization argument relative to the affine Weyl arrangement, we establish periodic alternant estimates for regular integral weights and corresponding rootwise bounds for all mixed radial derivatives of irreducible characters of compact connected semisimple groups.

Let \(\mathcal L\) be the Heisenberg-Virasoro type Lie superalgebra whose even part is the twisted Heisenberg-Virasoro algebra and whose odd part is an abelian ideal. We classify \(\mathbb Z_2\)-graded \(\mathcal L\)-modules that are free of super-rank \(1|1\) over \(U(\mathfrak h)=\mathbb C[L_0,F_0]\). Using polynomial shift operators, we express the defining relations as functional equations for the matrix entries of the generators. Assuming the rank one classification for the even subalgebra, we show that, up to parity reversal, every module with nontrivial odd action belongs to a triangular family \(\Omega^+(\lambda,\beta,d)\), whereas every module with trivial odd action is a direct sum of two rank-one modules for the even subalgebra. We determine the graded isomorphism classes: within the triangular family, the coupling polynomial \(d\) is invariant up to multiplication by a non-zero scalar. We also describe canonical proper submodules and show that every module in the classification is reducible. In particular, we provide a new realization of the Heisenberg-Virasoro type Lie superalgebra using $2\times2$ matrices with entries associated with polynomial shift-differential operators.

We study braid representations obtained from character-twisted Burau inputs by the Katz-Long-Moody (KLM) construction, which corresponds to multiplicative middle convolution for KZ-type equations. For $n\ge2$, let $q=e^{2\pi ia}$, $\tau=e^{2\pi ic}$, where $a,c>0$ and $nc+(n+1)a<1$. Known form and signature results give a positive-definite invariant form for $\lambda=e^{2\pi il}$, $0<l<1-nc-(n+1)a$. For fixed input, unitary realizations are unique up to simultaneous unitary conjugation. We address two questions. First, we determine spectra and invariant summands of the $M=n(n+1)$-dimensional output. The generator spectrum recovers $(q,\tau)$. The output decomposes into two trivial and two reduced Burau summands, and an $n(n-1)$-dimensional summand isomorphic to Long's reduced construction, generically irreducible for $n=3,4$. If $a$ or $c$ is irrational, every generator is non-Clifford in every finite-dimensional Weyl-Heisenberg frame; if $1,a,c$ are linearly independent over $\mathbb{Q}$, every generator is entangling for every nontrivial bipartition. The full braid image is not projectively dense. For general positive-definite unitary inputs, irreducibility of the full braid output at a nonexceptional parameter forces the free-group input operators to commute. Second, we construct one orthonormal basis giving each generator one block of size $2(n+1)$, $n-2$ blocks of size two, and identity elsewhere. Each generator is a product of $2n$ rank-one phase reflections and, in a fixed binary encoding, an $O(M)$-length word in one-qubit unitaries and CNOTs, without auxiliary qubits and including direction preparation and its inverse. We give finite-alphabet approximation, conditioning and error bounds, and a matched-resource comparison with a Quantum Shannon reference compiler. The gate bound is exponential in the encoded qubit count and excludes classical preprocessing.

Guo and Igusa introduced the derived and sub-derived delooping levels as refinements of the delooping level and asked how these two invariants compare, whether the derived delooping level always coincides with the opposite big finitistic dimension, and, if not, how large their difference can be. We answer these questions. First, we prove that every Artin algebra $A$ satisfies $\operatorname{ddell} A\leq \operatorname{sub-ddell} A$. This gives a positive answer to the ordinary-degree comparison problem, whereas a single explicit eight-dimensional algebra $C$ satisfies $k\text{-}\operatorname{sub-ddell} C=0<1=k\text{-}\operatorname{ddell} C$ for $k\geq 2$. We then compute, for every $k\geq 1$, all $k$-derived and $k$-sub-derived delooping levels of the monomial family of Barrios–Lanzilotta–Mata. In particular, that family satisfies $k\text{-}\operatorname{ddell} A_{n,s}=\operatorname{Findim}(A_{n,s}^{\mathrm{op}})$ for $k\geq 1$. In ordinary degree, this recovers an equality previously announced by Lam. We also construct a two-vertex, eleven-dimensional monomial algebra $\Lambda$, over an arbitrary field, such that $\operatorname{Findim}(\Lambda^{\mathrm{op}})=1<2=\operatorname{ddell}\Lambda=\operatorname{dell}\Lambda$. The same strict inequality is obtained by Gao, Liu and Xu using different algebras. A ten-dimensional non-monomial variant with a commutative corner algebra is also given. Finally, we construct a family of finite-dimensional monomial algebras $B_r$ satisfying $\operatorname{Findim}(B_r^{\mathrm{op}})=1$ and $\operatorname{ddell} B_r=2r+1$. Thus the difference $\operatorname{ddell} A-\operatorname{Findim}(A^{\mathrm{op}})$ is unbounded even among monomial algebras with fixed opposite big finitistic dimension.

For any finite-dimensional algebra $A$ over an algebraically closed field, we prove that if $A$ is $E$-infinite, then the Second brick-Brauer-Thrall Conjecture holds for $A$. In fact, we show that if there is a rational ray outside the $\tau$-tilting fan of $A$, then $A$ admits an infinite family of $\theta$-stable bricks of dimension $d$, for some positive integer $d$ and weight $\theta$. To show our main result, from any $\tau$-regular component of $A$ with no dense orbit, we obtain a brick component that also has no dense orbit and whose points in general position are $\theta$-stable modules for a given weight $\theta$. As an important consequence, we conclude that $A$ is stably-discrete (also called multiplicity-free) if and only if $A$ is $E$-finite. Moreover, we prove that Demonet's (lattice point) Conjecture implies the stable Second brick-Brauer-Thrall Conjecture. For $E$-infinite algebras, our results settle several open conjectures. Together with our earlier work and recent developments, this leads to important reductions in the study of some challenging open problems.

We establish a Chouinard-type theorem for modules over algebras of cochains on the classifying spaces of a broad class of infinite groups, allowing commutative ring spectra as coefficients. We then use this result to study finiteness properties of the integral cohomology rings of certain infinite groups, as well as stratification of the corresponding module categories. In particular, we complete the (co)stratification of module categories associated to homotopical groups over fields of positive characteristic.

We introduce the categories $\CM(X)$ and $\SCM(X)$ of reflexive sheaves on a minimal partial resolution $f\colon X \to \Spec R$ of a rational surface singularity $\Spec R$. The main result of this paper establishes that $\SCM(X)$ possesses a natural Frobenius structure, serving as a geometric counterpart to the algebraic Frobenius structure on special Cohen–Macaulay $R$-modules, introduced by Iyama–Wemyss and Iyama–Kalck–Wemyss–Yang. Utilizing this geometric framework, we establish an exact equivalence between $\SCM(X)$ and the category of special Cohen–Macaulay $R$-modules equipped with a specific exact structure, which induces a triangle equivalence between their stable categories. Consequently, this provides a direct, geometric proof of the Iyama–Kalck–Wemyss–Yang equivalence and yields a Buchweitz-type equivalence $\underline{\SCM}(X) \simeq D_{\sg}(X)$.

The para-Krawtchouk polynomials arose in the search for spin chains with perfect state transfer; the para-Racah, $q$-para-Racah and para-Bannai–Ito polynomials followed, obtained through singular truncations of families of the Askey scheme and used in turn to design spin chains. All are orthogonal on bi-lattices; the prefix goes back to the para-Krawtchouk case, whose spectrum is that of the parabose oscillator. The representations that these polynomials provide of the corresponding algebras — Hahn, Racah, $q$-Racah, Bannai–Ito and complementary Bannai–Ito — are shown to be reducible. Each decomposes into a direct sum of two irreducible modules of the same algebra, one attached to each of the two interlaced sub-lattices, and the parameters of the submodules are given. The para polynomials are thereby expressed in terms of the classical polynomials that the submodules carry, and the pairing $\lambda_n=\lambda_{N-n}$ of their eigenvalues is accounted for; the three features of the bi-lattice setting on which the reduction rests are identified. The properties of the para families are then presented in a unified way in the light of this decomposition, together with the constructions through which these families arise and the settings in which they have been put to use.

This paper considers the generic behaviour of sheaves supported on Schubert varieties in finite flag varieties. Our main results are that, for any fixed prime $p$, as the rank of the associated algebraic group grows, almost every intersection cohomology sheaf admits $p$-torsion in its stalks or costalks, and almost every indecomposable parity sheaf is not perverse. For certain families of partial flag varieties we also determine explicit asymptotics for this growth.

We prove the Fraser–Sherman-Bennett conjecture on the quasi-equivalence of positroid cluster structures arising from admissibly relabeled plabic graphs. We construct a tilting module from the boundary necklace and obtain a Frobenius categorification of the relabeled cluster structure. By identifying localized characters of rank-one modules with Plücker coordinates, we match the stable images of the mutable face modules with the summands of a reachable cluster-tilting object in the stable category. The induced derived equivalence yields a unimodular transformation of relative index lattices intertwining the full extended exchange matrices. These comparisons show that the identity map on the coordinate ring is a quasi-cluster isomorphism between the two cluster structures.

We consider generalised Casimir operators associated with irreducible representations for finite-dimensional, semisimple, complex Lie algebras. Using Klimyk's formula and Okubo's formula, we derive a rational ordinary generating function for the Casimir operator eigenvalues for simple Lie algebras which does not rely on knowledge of the Weyl group. We also directly reproduce generating function expressions for the classical Lie algebras originally due to Perelomov and Popov.

A Demazure model of an algebraic $k$-torus $T$ split by a finite Galois extension of fields $L/k$ and splitting group $G=\text{Gal}(L/k)$ is a smooth projective $T_L$-toric variety $X_{\Sigma}$ such that the natural action of $G$ on $T_L$ extends to $X_{\Sigma}$. A smooth projective model of the algebraic $k$-torus $T$ can then be taken to be the quotient of a Demazure model by its splitting group. The existence of Demazure models of algebraic $k$-tori was determined by Brylinski and further refined by Colliot-Thélène, Harari and Skorobogatov. Voskresenskii and Kunyavskii's birational classifications of algebraic $k$-tori in dimensions 2 and 3 involved constructions of Demazure models of the algebraic $k$-tori corresponding to maximal finite subgroups of $\text{GL}(r,\mathbb{Z})$ for $r=2,3$. We discuss some explicit constructions of Demazure models of algebraic $k$-tori, making connections with the defining integral representations of their splitting groups. The constructions determine smooth projective Demazure models of the algebraic tori corresponding to maximal finite subgroups of $\text{GL}(4,\mathbb{Z})$. Since projective toric varieties are determined by lattice polytopes, our constructions focus on some highly symmetric families of polytopes, such as root polytopes and central transportation polytopes.