Maximal Rank Research Articles

Reproducibility of mass spectrometry based metabolomics data

BackgroundAssessing the reproducibility of measurements is an important first step for improving the reliability of downstream analyses of high-throughput metabolomics experiments. We define a metabolite to be reproducible when it demonstrates consistency across replicate experiments. Similarly, metabolites which are not consistent across replicates can be labeled as irreproducible. In this work, we introduce and evaluate the use (Ma)ximum (R)ank (R)eproducibility (MaRR) to examine reproducibility in mass spectrometry-based metabolomics experiments. We examine reproducibility across technical or biological samples in three different mass spectrometry metabolomics (MS-Metabolomics) data sets.ResultsWe apply MaRR, a nonparametric approach that detects the change from reproducible to irreproducible signals using a maximal rank statistic. The advantage of using MaRR over model-based methods that it does not make parametric assumptions on the underlying distributions or dependence structures of reproducible metabolites. Using three MS Metabolomics data sets generated in the multi-center Genetic Epidemiology of Chronic Obstructive Pulmonary Disease (COPD) study, we applied the MaRR procedure after data processing to explore reproducibility across technical or biological samples. Under realistic settings of MS-Metabolomics data, the MaRR procedure effectively controls the False Discovery Rate (FDR) when there was a gradual reduction in correlation between replicate pairs for less highly ranked signals. Simulation studies also show that the MaRR procedure tends to have high power for detecting reproducible metabolites in most situations except for smaller values of proportion of reproducible metabolites. Bias (i.e., the difference between the estimated and the true value of reproducible signal proportions) values for simulations are also close to zero. The results reported from the real data show a higher level of reproducibility for technical replicates compared to biological replicates across all the three different datasets. In summary, we demonstrate that the MaRR procedure application can be adapted to various experimental designs, and that the nonparametric approach performs consistently well.ConclusionsThis research was motivated by reproducibility, which has proven to be a major obstacle in the use of genomic findings to advance clinical practice. In this paper, we developed a data-driven approach to assess the reproducibility of MS-Metabolomics data sets. The methods described in this paper are implemented in the open-source R package marr, which is freely available from Bioconductor at http://bioconductor.org/packages/marr.

Torus actions, maximality, and non-negative curvature

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

The Method of Operational Planning of Group Actions of Aircraft in the "Air Taxi" Mode

The article is devoted to the development of algorithms for operational planning of routes for a group of aircraft (AC). We consider group actions of small and unmanned aircraft in the "air taxi" mode, when there is no regular flight schedule between the points of destination, and requests are received "on call" for flights to points whose composition is unknown in advance and is of a random nature. The multicriteria task of planning a group flight in the "air taxi" mode is being solved. The solution to this problem is proposed using the apparatus of the queuing theory, according to which the system under consideration belongs to the class of multichannel queuing systems with waiting. A method for solving the problem of operational planning of aircraft actions is proposed. An algorithm for group target distribution of new claims between aircraft is developed on the basis of a modified minimax criterion for assigning the nearest aircraft for an object with a maximum service time. The developed algorithm is based on the following four main operations: in the first operation, priority unserved targets are selected according to the criterion of assigning a dynamic priority; in the second operation, the formed list is ranked according to another criterion, taking into account the importance and total distance of each ground object from the aircraft group, in three this operation selects the object with the maximum rank, and for it the task of assigning "own" aircraft is solved according to the third criterion of maximum proximity, in the fourth operation the conditions of non-intersection of the group flight routes are checked. A computer model of the system for servicing requests in the air taxi mode has been developed. The developed model makes it possible to analyze various scheduling algorithms, as well as to determine at each step the number of free claims and the number of free and busy aircraft. A comparison is made between the well-known in the theory of queuing and the proposed minimax approaches. It is shown that, in comparison with the known variants of scheduling in the queuing theory, on the basis of the proposed approach, the optimal number of used aircraft is achieved.

Equivalence, group of automorphism and invariants of a family of rank metric codes arising from linearized polynomials

Let Fq denote the finite field with q=pλ elements. Maximum Rank metric codes (MRD for short) are subsets of Mm×n(Fq) whose number of elements attains the Singleton-like bound. The first MRD codes known were found by Delsarte (1978) and Gabidulin (1985). Sheekey (2015) presented a new class of MRD codes over Fq called twisted Gabidulin codes and also proposed a generalization of the twisted Gabidulin codes to the codes Hk,s(L1,L2). The equivalence and duality of twisted Gabidulin codes were discussed by Lunardon, Trombetti, and Zhou (2016). A new class of MRD codes in M2n×2n(Fq) was found by Trombetti-Zhou (2018). In this work, we characterize the equivalence of the class of codes proposed by Sheekey, generalizing the results known for twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the paper, we restrict ourselves to the case L1(x)=x, where we present its right nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section, we present the automorphism group of Hk,s(x,L(x)) and compute its cardinality. In particular, we obtain the number of elements in the automorphism group of some twisted Gabidulin codes.

Energy and environmental efficiency of China’s transportation sectors considering CO2 emission uncertainty

• Energy and environmental efficiency of China’s transportation sector is investigated. • Uncertainty in estimation of CO 2 emission is considered. • Stochastic multicriteria acceptability analysis and data envelopment analysis are used. • Results show that great efficiency disparities exist among regions. • Uneven development has occurred in areas of China’s transportation sectors. China’s transportation sector suffers from energy over-consumption and CO 2 over-emission, resulting in increasing pressure to improve energy and environmental efficiency. Current measurement techniques cannot produce precise CO 2 emission data, and this uncertainty makes previous approaches problematic for analyzing energy and environmental efficiency. This study combines stochastic multicriteria acceptability analysis (SMAA-2) with data envelopment analysis (DEA) to evaluate the energy and environmental efficiency of Chinese transportation sectors in the presence of uncertain CO 2 emission data. The improved SMAA-DEA approach effectively handles CO 2 data uncertainty and also considers all possible input and output weights, thus providing meaningful information (such as maximum efficiency, average efficiency, and rank acceptability index) to guide the development of effective policies to improve efficiency. This study’s empirical findings show that the energy and environmental efficiency of transportation sectors in 30 provincial regions is poor, great efficiency disparities exist between regions, and uneven development has occurred in China.

Are Absolute Monocyte Count and The Lymphocyte-Monocyte Ratio (LMR) Valid Prognostic Factors In Classical Hodgkin Lymphoma? An Evaluation Based On Data From 1079 Patients

IntroductionThere is an increasing amount of data showing that both host immunity and tumor microenvironment play an essential role in determining the clinical course in patients with Hodgkin lymphoma (HL) and may also influence their response to therapy. Some investigators have considered the absolute lymphocyte count (ALC) as a surrogate bio-marker of tumor-infiltrating lymphocytes reflecting systemic host immunity, while the absolute monocyte count (AMC) is regarded as a surrogate bio-marker of tumor-associated macrophages reflecting the tumor microenvironment. For many years lymphopenia has been considered as a negative factor predicting poor outcome in HL. Recently monocytosis and the lymphocyte-monocyte ratio (LMR= ALC/AMC), which combines these two surrogate bio-markers, have also been shown to be useful prognostic parameters in HL, particularly in terms of progression free survival (PFS). AimsThe aim of the present study was to verify whether AMC and LMR are indeed valid prognostic parameters in HL, utilizing a large cohort of patients with the disease. Materials and MethodsWe collected data from 1079 patients with classical Hodgkin's lymphoma (cHL) enrolled in different clinical trials of “Gruppo Italiano Studio Linfomi (GISL)” during 1988-2007. Previously we had evaluated a database of 371 patients with cHL and examined different AMC and LMR cut-off levels using the maximum log rank test and c-Harell (area under the curve). We identified AMC of 600/mm3 and LMR of 3.5, as optimal cut-off values. ResultsPatients in our current study had the following characteristics: median age 32 years (range 17-71), 51% males, 15% stage IV disease, 74% with International Prognostic Score (IPS) 0-2, 72% nodular sclerosis and 18% mixed cellularity; 43% were treated with ABVD and 29% with BEACOPP/BEACOPP-like regimens. Radiotherapy (RT) was administered to 66% of cases. Median follow-up was 6.8 years (range 0.1-19), and 5-years and 10-years overall survival (OS) was 89 and 83%, respectively; 5-years PFS was 79%. In an earlier evaluation the 10-years PFS for patients with AMC >600/mm3 and LMR <3.5 was significantly worse, compared to the group with AMC <600/mm3 and LMR >3.5 (p-value: 0.032 and 0.005). However now after applying the same cut-off values to the larger database, we found no statistical differences in PFS ,and both threshold values for AMC (600/mm3) and LMR (3.5) lost their prognostic power. Using the same statistical methods for the cohort of 1079 patients, we found different cut-off values for AMC and LMR, that were equally effective in splitting different survival patterns. For AMC the value of 750/mm3 (26% of patients with AMC >750/mm3), and for LMR a ratio of 1.5 (19% of patients with LMR <1.5) revealed a difference in PFS which was statistically significant (p-value 0.003 and 0.001, respectively), as shown in Figure 1. In multivariate analysis after adjusting by IPS, RT and type of chemotherapy regimen administered, AMC and LMR maintained their prognostic significance for PFS (HR 1.37, p 0.022 and 1.37, p 0.037). After interaction with IPS (Figure 2) LMR showed an uniform pattern within the two risk -groups (overall: HR=1.42, p 0.019; IPS 0-2: HR=1.34, p 0.181; IPS 3-7: HR=1.49, p 0.048). [Display omitted] [Display omitted] ConclusionsOur study of 1079 patients confirms that peripheral monocyte and lymphocyte counts have an important prognostic impact in cHL. The cut off-values we used, although different from those utilized previously, were able to distinguish patients with different PFS. The cut-offs reported in the literature vary according to the different authors and the various sets of patients examined. At present we feel there is still a clear need to finally establish definitive and more appropriate cut-off levels that can be used for all future studies. In order to achieve this goal, it is essential to study an even larger series of patients within the framework of an international collaborative study. Disclosures:No relevant conflicts of interest to declare.

Compact Kähler threefolds with the action of an abelian group of maximal rank

In this note, we study the normal compact K\"ahler (possibly singular) threefold $X$ admitting the action of a free abelian group $G$ of maximal rank, all the non-trivial elements of which are of positive entropy. If such $X$ is further assumed to have only terminal singularities, then we prove that it is either a rationally connected projective threefold or bimeromorphic to a quasi-\'etale quotient of a complex $3$-torus.

Obstructions to Semiorthogonal Decompositions for Singular Threefolds I: K-Theory

We investigate necessary conditions for Gorenstein projective varieties to admit semiorthogonal decompositions introduced by Kawamata, with main emphasis on threefolds with isolated compound $A_n$ singularities. We introduce obstructions coming from Algebraic $\mathrm{K}$-theory and translate them into the concept of maximal nonfactoriality. Using these obstructions we show that many classes of nodal threefolds do not admit Kawamata type semiorthogonal decompositions. These include nodal hypersurfaces and double solids, with the exception of a nodal quadric, and del Pezzo threefolds of degrees $1 \le d \le 4$ with maximal class group rank. We also investigate when does a blow up of a smooth threefold in a singular curve admit a Kawamata type semiorthogonal decomposition and we give a complete answer to this question when the curve is nodal and has only rational components.

A combinatorial algorithm for computing the rank of a generic partitioned matrix with $$2 \times 2$$ submatrices

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) $A = (A_{\alpha \beta} x_{\alpha \beta})$, where $A_{\alpha \beta}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$ and $x_{\alpha \beta}$ is an indeterminate for $\alpha = 1,2,\dots, \mu$ and $\beta = 1,2, \dots, \nu$. This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos, Qiao, and Subrahamanyam (2018) and Garg, Gurvits, Oliveiva, and Wigderson (2019), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial $O((\mu \nu)^2 \min \{ \mu, \nu \})$-time algorithm for computing the symbolic rank of a $(2 \times 2)$-type generic partitioned matrix of size $2\mu \times 2\nu$. Our algorithm is inspired by the Wong sequence algorithm by Ivanyos, Qiao, and Subrahamanyam for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of $A$ for an arbitrary field $\mathbf{F}$.

First-order invariants of differential 2-forms

Let M be a smooth manifold of dimension 2n, and let OM be the dense open subbundle in ∧2 T*M of 2-covectors of maximal rank. The algebra of Diff M -invariant smooth functions of first order on OM is proved to be isomorphic to the algebra of smooth Sp(Ωx)-invariant functions on , x being a fixed point in M , and Ωx a fixed element in (OM ) x . The maximum number of functionally independent invariants is computed.

Evaluating the added value of multi-input atmospheric transport ensemble modeling for applications of the Comprehensive Nuclear Test-Ban Treaty organization (CTBTO)

The Comprehensive Nuclear Test-Ban Treaty Organization (CTBTO) runs to date operationally an atmospheric transport modeling chain in backward mode based on operational deterministic European Centre for Medium-Range Weather Forecasts-Integrated Forecasting System (ECMWF-IFS) and on National Centers for Environmental Prediction-Global Forecast System (NCEP-GFS) input data. Meanwhile, ensemble dispersion modeling is becoming more and more widespread due to the ever increasing computational power and storage capacities. The potential benefit of this approach for current and possible future CTBTO applications was investigated using data from the ECMWF-Ensemble Prediction System (EPS). Five different test cases - among which are the ETEX-I experiment and the Fukushima accident - were run in backward or forward mode and - in the light of a future operational application - special emphasis was put on the performance of an arbitrarily selected 10- versus the full 51-member ensemble. For those test cases run in backward mode and based on a puff release it became evident that Possible Source Regions (PSRs) can be meaningfully reduced in size compared to results based solely on the deterministic run by applying minimum and probability of exceedance ensemble metrics. It was further demonstrated that a given puff release of 4E10 Bq of Se-75 can be reproduced within the meteorological uncertainty range [1.9E9 Bq,1.7E13 Bq] including a probability for not exceeding an assumed upper limit source term using simple scaling of a measurement with the corresponding ensemble metrics of backward fields. For the test cases run in forward mode it was found that the control run as well as 10- and 51-member medians all exhibit similar performance in time series evaluation. Maximum rank difference adds up to less than 10% with reference to possible rank values [0,4]. The maximum difference in the Brier score for both ensembles is less than 3%. The main added value of the ensemble lies in producing meteorologically induced concentration uncertainties and thus explaining observed measurements at specific sites. Depending on the specific test case and on the ensemble size between 27 and 74% of samples all lie within concentration ranges derived from the different meteorological fields used. In the future uncertainty information per sample could be used in a full source term inversion to account for the meteorological uncertainty in a proper way. It can be concluded that a 10-member meteorological ensemble is good enough to already benefit from useful ensemble properties. Meteorological uncertainty to a large degree is covered by the 10-member subset because forecast uncertainty is largely suppressed due to concatenating analyses and short term forecasts, as required in the operational CTBTO procedure, on which this study focuses. Besides, members from different analyses times are on average unrelated. It was recommended to Working Group B of CTBTO to implement the ensemble system software in the near future.

Eulerianity of Fourier coefficients of automorphic forms

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a ‘hidden’ invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier–Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.

A rank question for homogeneous polynomials

A number of open problems in the field of several complex variables naturally lead to the study of bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) on C n + 1 \mathbb {C}^{n+1} . In particular, both the Ebenfelt sum of squares conjecture and the degree estimate conjecture for proper rational mappings between balls in complex Euclidean spaces lead to the study of the rank of the bihomogeneous polynomial r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z}\right \rVert ^2 under certain additional hypotheses. When r r has a diagonal coefficient matrix, these questions reduce to questions about real homogeneous polynomials. More specifically, we are led to study the rank of P = S Q P=SQ when Q Q is a homogeneous polynomial and S ( x ) = ∑ j = 0 n x j S(x) = \sum _{j=0}^n x_j . In this paper, we use techniques from commutative algebra to estimate the minimum rank of P = S Q P=SQ under the additional hypothesis that Q Q has maximum rank. The problem has already been solved for n + 1 ≤ 3 n+1 \leq 3 , and so we consider n + 1 ≥ 4 n+1\geq 4 . We obtain a minimum rank estimate that is sharp when n + 1 = 4 n+1=4 , and we exhibit a family of polynomials having this minimum rank. We also prove an estimate for n + 1 &gt; 4 n+1&gt;4 that, while not sharp, is non-trivial.

Doubling nodal solutions to the Yamabe equation in [formula omitted] with maximal rank

We construct a new family of entire solutions to the Yamabe equation−Δu=n(n−2)4|u|4n−2u in D1,2(Rn). If n=3 our solutions have maximal rank, being the first example in odd dimension. Our construction has analogies with the doubling of the equatorial spheres in the construction of minimal surfaces in S3(1).

Scattered subspaces and related codes

After a seminal paper by Shekeey (Adv Math Commun 10(3):475-488, 2016), a connection between maximum h-scattered $${{\mathbb {F}}}_{q}$$ -subspaces of $$V(r,q^n)$$ and maximum rank distance (MRD) codes has been established in the extremal cases $$h=1$$ and $$h=r-1$$ . In this paper, we propose a connection for any $$h\in \{1,\ldots ,r-1\}$$ , extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. We show that, up to equivalence, MRD codes having the same parameters as the ones in our connection come from an h-scattered subspace. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum h-scattered subspaces.

Isotropic Quot schemes of orthogonal bundles over a curve

We study the isotropic Quot schemes [Formula: see text] parametrizing degree [Formula: see text] isotropic subsheaves of maximal rank of an orthogonal bundle [Formula: see text] over a curve. The scheme [Formula: see text] contains a compactification of the space [Formula: see text] of degree [Formula: see text] maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [D. Cheong, I. Choe and G. H. Hitching, Irreducibility of Lagrangian Quot schemes over an algebraic curve, preprint (2019), arXiv:1804.00052, v2]. We observe that for certain topological types of [Formula: see text], the scheme [Formula: see text] is empty for all [Formula: see text]. In the remaining cases, for infinitely many [Formula: see text] there are irreducible components of [Formula: see text] consisting entirely of nonsaturated subsheaves, and so [Formula: see text] is strictly larger than the closure of [Formula: see text]. As our main result, we prove that for any orthogonal bundle [Formula: see text] and for [Formula: see text], the closure [Formula: see text] of [Formula: see text] is either empty or consists of one or two irreducible connected components, depending on [Formula: see text] and [Formula: see text]. In so doing, we also characterize the nonsaturated part of [Formula: see text] when [Formula: see text] has even rank.

Local Kalman rank condition for linear time varying systems

&lt;p style='text-indent:20px;'&gt;In this paper, we study some non-negative integers related to a linear time varying system and to some Krylov sub-spaces associated to this system. Such integers are similar to the controllability indices and have been used in the literature to derive results on the controllability of linear systems. The purpose of this paper goes in the same direction by studying the local behavior of these integers especially nearby instants in the time interval with some maximal rank condition and then apply them to get some results which generalize the mentioned existing results.&lt;/p&gt;

Exact computation of maximum rank correlation estimator

SummaryIn this paper we provide a computation algorithm to get a global solution for the maximum rank correlation estimator using the mixed integer programming (MIP) approach. We construct a new constrained optimization problem by transforming all indicator functions into binary parameters to be estimated and show that it is equivalent to the original problem. We also consider an application of the best subset rank prediction and show that the original optimization problem can be reformulated as MIP. We derive the nonasymptotic bound for the tail probability of the predictive performance measure. We investigate the performance of the MIP algorithm by an empirical example and Monte Carlo simulations.

Quasiflats in hierarchically hyperbolic spaces

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Several noteworthy examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups; the maximal rank of a free abelian subgroup for right-angled Coxeter groups and right-angled Artin groups (in the latter this can also be observed as the clique number of the defining graph); and, for the Weil–Petersson metric, the rank is the integer part of half the complex dimension of Teichmüller space. We prove that, in a hierarchically hyperbolic space (HHS), any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a very mild condition on the HHS satisfied by all natural examples). This resolves outstanding conjectures when applied to a number of different groups and spaces. In the case of the mapping class group, we verify a conjecture of Farb. For Teichmüller space we answer a question of Brock. In the context of certain CAT(0) cubical groups, our result handles novel special cases, including right-angled Coxeter groups. An important ingredient in the proof, which we expect will have other applications, is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. (If the HHS is a CAT(0) cube complex, then the rank can be lower than the dimension of the space.) We deduce a number of applications of these results. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain factored spaces, which are simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups. Another application of our results is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. As a template, we give a new proof of quasi-isometric rigidity of mapping class groups, which, once we have established our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.

Rank-deficient representations in the theta correspondence over finite fields arise from quantum codes

Let V V be a symplectic vector space and let μ \mu be the oscillator representation of Sp ⁡ ( V ) \operatorname {Sp}(V) . It is natural to ask how the tensor power representation μ ⊗ t \mu ^{\otimes t} decomposes. If V V is a real vector space, then the theta correspondence asserts that there is a one-one correspondence between the irreducible subrepresentations of Sp ⁡ ( V ) \operatorname {Sp}(V) and the irreps of an orthogonal group O ( t ) O(t) . It is well-known that this duality fails over finite fields. Addressing this situation, Gurevich and Howe have recently assigned a notion of rank to each Sp ⁡ ( V ) \operatorname {Sp}(V) representation. They show that a variant of the Theta correspondence continues to hold over finite fields, if one restricts attention to subrepresentations of maximal rank. The nature of the rank-deficient components was left open. Here, we show that all rank-deficient Sp ⁡ ( V ) \operatorname {Sp}(V) -subrepresentations arise from embeddings of lower-order tensor products of μ \mu and μ ¯ \bar \mu into μ ⊗ t \mu ^{\otimes t} . The embeddings live on spaces that have been studied in quantum information theory as tensor powers of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes. We then find that the irreducible Sp ⁡ ( V ) \operatorname {Sp}(V) -subrepresentations of μ ⊗ t \mu ^{\otimes t} are labelled by the irreps of orthogonal groups O ( r ) O(r) acting on certain r r -dimensional spaces for r ≤ t r\leq t . The results hold in odd charachteristic and the “stable range” t ≤ 1 2 dim ⁡ V t\leq \frac 12 \dim V . Our work has implications for the representation theory of the Clifford group. It can be thought of as a generalization of the known characterization of the invariants of the Clifford group in terms of self-dual codes.