Recent zbMATH articles in MSC 20Chttps://www.zbmath.org/atom/cc/20C2021-04-16T16:22:00+00:00WerkzeugThe homogeneous pseudo-embeddings and hyperovals of the generalized quadrangle \(H(3,4)\).https://www.zbmath.org/1456.510032021-04-16T16:22:00+00:00"De Bruyn, Bart"https://www.zbmath.org/authors/?q=ai:de-bruyn.bart"Gao, Mou"https://www.zbmath.org/authors/?q=ai:gao.mouLet \(\mathcal{S}=(\mathcal{P},\mathcal{L})\) be a generalized quadrangle and denote by \(\varepsilon:\mathcal{S} \longrightarrow \mathrm{PG}(V)\) the universal pseudo-embedding \(\mathcal{S}.\) The vector dimension of the subspace \(\langle \varepsilon(\mathcal{P})\rangle\) of \(V\) is called the \textit{pseudo-embedding rank} of \(\mathcal{S}.\)
If \(|\mathcal{P}|\) is finite, denote by \(C=C(\mathcal{S})\) the binary code of length \(|\mathcal{P}|\) generated by the characteristic vectors of the lines of \(\mathcal{S}.\) It has been proved in [the first author, Adv. Geom. 13, No. 1, 71--95 (2013; Zbl 1267.51002)] that the pseudo-embedding rank of \(\mathcal{S}\) is \(|\mathcal{P}|- \dim(C).\)
In this paper, the authors prove that the pseudo-embedding rank of the Hermitian quadrangle \(H(3,4)\) is equal to \(24.\) As a consequence, the binary code \(C(H(3,4))\) has dimension \(45-24=21,\) because the generalized quadrangle \(H(3,4)\) has \(45\) points.
They also show that there are, up to isomorphism, four homogeneous pseudo-embeddings of \(H(3,4),\) with respective vector dimensions \(14,\) \(15,\) \(23\) and \(24.\)
Reviewer: Guglielmo Lunardon (Napoli)From the monster to Thompson to O'Nan.https://www.zbmath.org/1456.110632021-04-16T16:22:00+00:00"Duncan, John F. R."https://www.zbmath.org/authors/?q=ai:duncan.john-f-rSummary: The commencement of monstrous moonshine is a connection between the largest sporadic simple group-the monster-and complex elliptic curves. Here we explain how a closer look at this connection leads, via the Thompson group, to recently observed relationships between the non-monstrous sporadic simple group of O'Nan and certain families of elliptic curves defined over the rationals. We also describe umbral moonshine from this perspective.
For the entire collection see [Zbl 1452.17002].Equivariant character correspondences and inductive McKay condition for type \(\mathsf{A}\).https://www.zbmath.org/1456.200092021-04-16T16:22:00+00:00"Cabanes, Marc"https://www.zbmath.org/authors/?q=ai:cabanes.marc"Späth, Britta"https://www.zbmath.org/authors/?q=ai:spath.brittaSummary: As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type \(\mathsf{A}_{n-1}\), split or twisted. Key to the proofs is the study of certain characters of \(\mathrm{SL}_n(q)\) and \(\mathrm{SU}_n(q)\) related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.On group rings and some of their applications to combinatorics and symmetric cryptography.https://www.zbmath.org/1456.940582021-04-16T16:22:00+00:00"Carlet, Claude"https://www.zbmath.org/authors/?q=ai:carlet.claude"Tan, Yin"https://www.zbmath.org/authors/?q=ai:tan.yinSummary: We give a survey of recent applications of group rings to combinatorics and to cryptography, including their use in the differential cryptanalysis of block ciphers.Sylow like theorems for \(V(\mathbb{Z}G)\).https://www.zbmath.org/1456.160372021-04-16T16:22:00+00:00"Kimmerle, Wolfgang"https://www.zbmath.org/authors/?q=ai:kimmerle.wolfgangSummary: The main part of this article is a survey on torsion subgroups of the unit group of an integral group ring. It contains the major parts of my talk given at the conference ``Groups, Group Rings and Related Topics'' at UAEU in Al Ain October 2013. In the second part special emphasis is layed on \(p\)-subgroups and on the open question whether there is a Sylow like theorem in the normalized unit group of an integral group ring. For specific classes of finite groups we prove that \(p\)-subgroups of the normalized unit group of its integral group rings \(V(\mathbb{Z}G)\) are isomorphic to subgroups of \(G \). In particular for \(p = 2\) this is shown when \(G\) has abelian Sylow 2-subgroups. This extends results known for soluble groups to classes of groups which are not contained in the class of soluble groups.Limits, standard complexes and \(\mathbf{fr} \)-codes.https://www.zbmath.org/1456.180022021-04-16T16:22:00+00:00"Ivanov, Sergeĭ O."https://www.zbmath.org/authors/?q=ai:ivanov.sergei-o"Mikhailov, Roman V."https://www.zbmath.org/authors/?q=ai:mikhailov.roman"Pavutnitskiy, Fedor Yu."https://www.zbmath.org/authors/?q=ai:pavutnitskiy.fedor-yuLet \(G\) be a group and denote by \(\mathrm{Pres}(G)\) the \textit{category of presentations} of \(G\), whose objects are surjective group morphisms from a free group to \(G\) and morphisms obvious commutative triangles. The article under review deals with nice subfunctors of the functor mapping \(F\twoheadrightarrow G\) (with \(F\) free) to the group ring \(\mathbb{Z}[F]\). One, denoted by \(\mathbf{f}\), is given by the augmentation ideal, and another, denoted by \(\mathbf{g}\), by the kernel of the surjective map \(\mathbb{Z}[F]\twoheadrightarrow\mathbb{Z}[G]\) induced by the presentation: \(\mathbf{f}\) and \(\mathbf{g}\) define functorial (two-sided) ideals of \(\mathbb{Z}[-]\), so that one can build a lot of other ones from them by taking products, intersections or sums. Such ideals are called \(\mathbf{fr}\)-\textit{codes}; for example, \(\mathbf{f}^2\mathbf{r}+(\mathbf{r}^2\cap\mathbf{frf})\) is an \(\mathbf{fr}\)-code. The authors study the problem that they call \textit{translation} of an \(\mathbf{fr}\)-code, that is the description of its higher limits viewed as a functor \(\mathrm{Pres}(G)\to\mathbf{Ab}\).
The article ends with several computational examples, as: \(\lim^i \mathbf{r^2+frf+rf^2}\), which is naturally isomorphic to the group homology \(H_2(G;G_{ab})\) for \(i=1\), to \(G_{ab}^{\otimes 2}\) for \(i=2\) , and to \(0\) for \(i>2\). This is obtained by some general --- elementary but nice -- results in functor cohomology.
Let \(\mathcal{C}\) be a small category and \(X : \mathcal{C}\to\mathbf{Ab}\) a functor. Assume that \(\mathcal{C}\) fulfils both following conditions:
-- for any objects \(c\) and \(d\) of \(\mathcal{C}\), the set \(\mathrm{Hom}(c,d)\) is non-empty;
-- every pair of objects in \(\mathcal{C}\) has a coproduct.
(But one does not require the category to have an initial object, what would imply that all higher limits over \(\mathcal{C}\) are zero. Note that \(\mathrm{Pres}(G)\) always satisfies these conditions.)
It is not hard to see that \(\underset{\mathcal{C}}{\lim}X\) identifies with the equalizer of both canonical arrows \(X(c)\to X(c\sqcup c)\) for any object \(c\) of \(\mathcal{C}\). In Section 2 of the article, the authors generalise as follows this fact: they construct a cosimplicial object in \(\mathcal{C}\) denoted by \(\mathbf{B}(c)\), which is natural in \(c\), such that \(\mathbf{B}(c)^n\) is the coproduct of \((n+1)\) copies of \(c\). They show that the homotopy type of \(\mathbf{B}(c)\) does not depend on \(c\) (\textit{Theorem 1}). Moreover, the cohomotopy of \(X\big(\mathbf{B}(c)\big)\) (meaning the cohomology of the cosimplicial complex given by alternate sum of cofaces, for example) is naturally isomorphic to higher limits of \(X\) (\textit{Theorem 2}).
The article introduces also a notion of (polynomial) degree for a functor \(\mathcal{C}\to\mathbf{Ab}\), which is very much reminiscent of [\textit{S. Eilenberg} and \textit{S. MacLane}, Ann. Math. (2) 60, 49--139 (1954; Zbl 0055.41704)] and, above all, [\textit{T. Pirashvili}, Tr. Tbilis. Mat. Inst. Razmadze 70, 69--91 (1982; Zbl 0521.18015)], and which allows to get vanishing of some higher limits in large cohomological degree (\textit{Proposition 3}). Before applying this to \(\mathbf{fr}\)-codes, the authors give also a Künneth-like formula (\textit{Theorem 4}) in functor cohomology (still in the setting of a source category satisfying the previous assumptions).
Reviewer: Aurelien Djament (Villeneuve d'Ascq)The conjugacy problem in \(\mathrm{ Gl } ( n, \mathbb{Z} )\).https://www.zbmath.org/1456.200042021-04-16T16:22:00+00:00"Eick, Bettina"https://www.zbmath.org/authors/?q=ai:eick.bettina"Hofmann, Tommy"https://www.zbmath.org/authors/?q=ai:hofmann.tommy"O'Brien, E. A."https://www.zbmath.org/authors/?q=ai:obrien.eamonn-aLet \(T\) and \(U\) be elements of \(\mathrm{GL}(n,\mathbb{Q})\). The \textit{rational conjugacy problem} asks if there exists \(X\in\mathrm{GL}(n, \mathbb{Q})\) such that \(XTX^{-1}=U\). It is well known that this can be decided effectively by computing and comparing the rational canonical forms of \(T\) and \(U\). More difficult is the \textit{integral conjugacy problem}: decide whether there exists \(X\in\mathrm{GL}(n,\mathbb{Z})\) with \(XTX^{-1}=U\). Associated to the integral conjugacy problem is the \textit{centraliser problem}: determine a generating set for \(C_{\mathbb{Z}}(T)=\{X\in\mathrm{GL}(n,\mathbb{Z}) \mid XTX^{-1}=T\}\). Since \(C_{\mathbb{Z}}(T)\) is arithmetic, the work of \textit{F. Grunewald} and \textit{D. Segal} [Ann. Math. (2) 112, 531--583 (1980; Zbl 0457.20047)] implies that it is both finitely generated and finitely presented. But no practical algorithm to compute a finite generating set for an arithmetic group is known.
In the article under review, the authors present a new algorithm that, given two matrices in \(\mathrm{GL}(n,\mathbb{Q})\), decides if they are conjugate in \(\mathrm{GL}(n, \mathbb{Z})\) and, if so, determines a conjugating matrix. They also give an algorithm to construct a generating set for \(C_{\mathbb{Z}}(T)\) for \(T \in\mathrm{GL}(n,\mathbb{Q})\). To do this they reduce these problems, respectively, to the isomorphism and automorphism group problems for certain modules over rings of the form \(\mathcal{O}_{K}[y]/(y^{\ell})\), where \(\mathcal{O}_{K}\) is the maximal order of an algebraic number field \(K\) and \(\ell \in \mathbb{N}\), and then provide algorithms to solve the latter. The algorithms are effective and usable in practice, the authors provide their implementation using the MAGMA software.
Reviewer: Enrico Jabara (Venezia)Automorphisms of types in certain type theories and representation of finite groups.https://www.zbmath.org/1456.030282021-04-16T16:22:00+00:00"Soloviev, Sergei"https://www.zbmath.org/authors/?q=ai:soloviev.sergei-vladimirovich|solovev.sergei-i|solovyev.sergey-alexandrovichSummary: The automorphism groups of types in several systems of type theory are studied. It is shown that in simply typed \(\lambda\)-calculus \(\lambda^1\beta\eta\) and in its extension with surjective pairing and terminal object these groups correspond exactly to the groups of automorphisms of finite trees. In second-order \(\lambda\)-calculus and in Luo's framework (LF) with dependent products, any finite group may be represented.Quasi-invariants in characteristic \(p\) and twisted quasi-invariants.https://www.zbmath.org/1456.812522021-04-16T16:22:00+00:00"Ren, Michael"https://www.zbmath.org/authors/?q=ai:ren.michael-s"Xu, Xiaomeng"https://www.zbmath.org/authors/?q=ai:xu.xiaomengSummary: The spaces of quasi-invariant polynomials were introduced by \textit{O. A. Chalykh} and \textit{A. P. Veselov} [Commun. Math. Phys.126, No. 3, 597-611 (1990; Zbl 0746.47025)]. Their Hilbert series over fields of characteristic 0 were computed by \textit{M. Feigin} and \textit{A. P. Veselov} [Int. Math. Res. Not. 2002, No. 10, 521-545 (2002; Zbl 1009.20044)]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, \textit{A. Braverman, P. Etingof}, and \textit{M. Finkelberg} [preprint (2020; \url{arxiv:1611.10216})] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.Algebras of generalized dihedral type.https://www.zbmath.org/1456.160082021-04-16T16:22:00+00:00"Erdmann, Karin"https://www.zbmath.org/authors/?q=ai:erdmann.karin"Skowroński, Andrzej"https://www.zbmath.org/authors/?q=ai:skowronski.andrzejSummary: We provide a complete classification of all algebras of generalized dihedral type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.The intrinsic hyperplane arrangement in an arbitrary irreducible representation of the symmetric group.https://www.zbmath.org/1456.200082021-04-16T16:22:00+00:00"Tsilevich, N. V."https://www.zbmath.org/authors/?q=ai:tsilevich.natalia-v"Vershik, A. M."https://www.zbmath.org/authors/?q=ai:vershik.anatoli-m"Yuzvinsky, S."https://www.zbmath.org/authors/?q=ai:yuzvinskij.s-aThe main result of the paper says that for every irreducible complex representation \(\pi_\lambda\) of the symmetric group \(S_n\) there exists a canonical ``intrinsic'' hyperplane arrangement \(A_\lambda\) in the space \(V_\lambda\) of this representation. In the case of the natural representation of \(S_n\),
it coincides with the so-called braid arrangement, studied by [\textit{V. I. Arnol'd}, Math. Notes 5, 138--140 (1969; Zbl 0277.55002); translation from Mat. Zametki 5, 227--231 (1969)] in connection
with the cohomology of the group of pure1 braids.
This arrangement has a natural description in terms of invariant subspaces of Young subgroups, and enjoys a number of remarkable properties. An attempt to generalize Arnold's construction to other irreducible representations of symmetric groups has led the authors to quite dissimilar arrangements,
whose complements, in particular, are not \(K(\pi,1)\) spaces.
Reviewer: Marek Golasiński (Olsztyn)Rationality problem for norm one tori in small dimensions.https://www.zbmath.org/1456.110432021-04-16T16:22:00+00:00"Hasegawa, Sumito"https://www.zbmath.org/authors/?q=ai:hasegawa.sumito"Hoshi, Akinari"https://www.zbmath.org/authors/?q=ai:hoshi.akinari"Yamasaki, Aiichi"https://www.zbmath.org/authors/?q=ai:yamasaki.aiichiSummary: We classify stably/retract rational norm one tori in dimension \(n-1\) for \(n=2^e (e\geq 1)\) as a power of \(2\) and \(n=12, 14, 15\). Retract non-rationality of norm one tori for primitive \(G\leq S_{2p}\) where \(p\) is a prime number and for the five Mathieu groups \(M_n\leq S_n (n=11,12,22,23,24)\) is also given.Landau's theorem for \(\pi\)-blocks of \(\pi\)-separable groups.https://www.zbmath.org/1456.200072021-04-16T16:22:00+00:00"Sambale, Benjamin"https://www.zbmath.org/authors/?q=ai:sambale.benjaminLet \(\pi\) be a set of primes. The main result of the paper shows that the order of a defect group of a \(\pi\)-block \(B\) of a \(\pi\)-separable group can be bounded by a function depending only on \(k(B)\), the number of irreducible characters in \(B\). This generalizes a result by the reviewer [J. Reine Angew. Math. 404, 189--191 (1990; Zbl 0684.20007)] on \(p\)-blocks of \(p\)-solvable groups where \(p\) is a single prime. The author also works out explicitly the cases where \(k(B) \le 3\).
Reviewer: Burkhard Külshammer (Jena)Color Lie rings and PBW deformations of skew group algebras.https://www.zbmath.org/1456.170162021-04-16T16:22:00+00:00"Fryer, S."https://www.zbmath.org/authors/?q=ai:fryer.sian"Kanstrup, T."https://www.zbmath.org/authors/?q=ai:kanstrup.tina"Kirkman, E."https://www.zbmath.org/authors/?q=ai:kirkman.ellen-e"Shepler, A. V."https://www.zbmath.org/authors/?q=ai:shepler.anne-v"Witherspoon, S."https://www.zbmath.org/authors/?q=ai:witherspoon.sarah-jThe authors study color Lie rings over finite group algebras and the corresponding universal enveloping algebras [\textit{M. Scheunert}, J. Math. Phys. 20, 712--720 (1979; Zbl 0423.17003)]. More precisely, they consider such rings that arise from finite abelian groups acting diagonally on a finite dimensional vector space over a field of characteristic 0 and prove that their universal enveloping algebras can be presented as quantum Drinfeld orbifold algebras [\textit{P. Shroff}, Commun. Algebra 43, No. 4, 1563--1570 (2015; Zbl 1332.16020)]. The proof is mainly based on the theory of PBW deformations and related tools (see, e.g. [\textit{P. Shroff} and \textit{S. Witherspoon}, J. Algebra Appl. 15, No. 3, Article ID 1650049, 15 p. (2016; Zbl 1345.16025)]. As an application they show that these algebras are braided Hopf algebras.
Reviewer: Aleksandr G. Aleksandrov (Moskva)Weil representations via abstract data and Heisenberg groups: a comparison.https://www.zbmath.org/1456.200022021-04-16T16:22:00+00:00"Cruickshank, J."https://www.zbmath.org/authors/?q=ai:cruickshank.james"Gutiérrez Frez, L."https://www.zbmath.org/authors/?q=ai:gutierrez-frez.luis"Szechtman, F."https://www.zbmath.org/authors/?q=ai:szechtman.fernandoThe paper provides Weil representations of unitary groups with even rank over finite rings via Heisenberg groups. The authors use a constructive approach to obtain the explicit matrix form of the Bruhat elements as well as information on generalized Gauss sums. The result is then shown to be identical to the one following from axiomatic considerations. When the ring is local (not necessarily finite) on the other hand, the index of the subgroup generated by the Bruhat elements is computed. Although the subject of the paper is rather technical, all concepts are explained clearly, results are layed down in great detail and proofs are given in a consistent rigorous manner. The authors also provide several examples at the end as well as a nice selection of references. In view of all this, the article might be interesting not only to specialists in the field, but also to graduate students, due to its pedagogical merits.
Reviewer: Danail Brezov (Sofia)Generalized stability of Heisenberg coefficients.https://www.zbmath.org/1456.051772021-04-16T16:22:00+00:00"Ying, Li"https://www.zbmath.org/authors/?q=ai:ying.liKronecker coefficients and Littlewood-Richardson coefficients are two famous structure constants, which can be defined in terms of representations of symmetric groups. Heisenberg coefficients are Schur structure constants of the Heisenberg product which generalize both Littlewood-Richardson coefficients and Kronecker coefficients. In this paper, the author shows that any stable triple of partitions for Kronecker coefficients or Littlewood-Richardson coefficients also stabilizes Heisenberg coefficients, and he classifies the triples stabilizing Heisenberg coefficients. Also, the author follows Vallejo's idea of using matrix additivity to generate Heisenberg stable triples.
Reviewer: Duško Bogdanić (Banja Luka)Large orbit sizes in finite group actions.https://www.zbmath.org/1456.200062021-04-16T16:22:00+00:00"Qian, Guohua"https://www.zbmath.org/authors/?q=ai:qian.guohua"Yang, Yong"https://www.zbmath.org/authors/?q=ai:yang.yongLet \(G\) be a finite group acting faithfully on a finite vector space \(V\). The \(G\)-orbit of an element \(v \in V\) is the set \(v^{G}=\{v^{g} \mid g \in G \}\) and the orbit size of \(v\) is \(| v^{G}|\). Results on orbit sizes, particularly on the existence of large orbits, have been fundamental to solving problems in several areas of finite group theory.
In the paper under review, the authors study relations of the sizes of various sections of finite linear groups and the largest orbit size of linear group actions. They also provide various applications of the results they have obtained.
Reviewer: Enrico Jabara (Venezia)Cylindric Hecke characters and Gromov-Witten invariants via the asymmetric six-vertex model.https://www.zbmath.org/1456.051742021-04-16T16:22:00+00:00"Korff, Christian"https://www.zbmath.org/authors/?q=ai:korff.christianSummary: We construct a family of infinite-dimensional positive sub-coalgebras within the Grothendieck ring of Hecke algebras, when viewed as a Hopf algebra with respect to the induction and restriction functor. These sub-coalgebras have as structure constants the 3-point genus zero Gromov-Witten invariants of Grassmannians and are spanned by what we call cylindric Hecke characters, a particular set of virtual characters for whose computation we give several explicit combinatorial formulae. One of these expressions is a generalisation of Ram's formula for irreducible Hecke characters and uses cylindric broken rim hook tableaux. We show that the latter are in bijection with so-called `ice configurations' on a cylindrical square lattice, which define the asymmetric six-vertex model in statistical mechanics. A key ingredient of our construction is an extension of the boson-fermion correspondence to Hecke algebras and employing the latter we find new expressions for Jing's vertex operators of Hall-Littlewood functions in terms of the six-vertex transfer matrices on the infinite planar lattice.Computing character degrees via a Galois connection.https://www.zbmath.org/1456.200052021-04-16T16:22:00+00:00"Lewis, Mark L."https://www.zbmath.org/authors/?q=ai:lewis.mark-l"McVey, John K."https://www.zbmath.org/authors/?q=ai:mcvey.john-kSummary: In a previous paper, the second author established that, given finite fields \(F < E\) and certain subgroups \(C\leq E^\times\), there is a Galois connection between the intermediate field lattice \(\{L \mid F \leq L \leq E\}\) and \(C\)'s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product \(C\rtimes\mathrm{Gal}(E/F)\). However, the analysis when \(|F|\) is a Mersenne prime is more complicated, so certain cases were omitted from that paper. The present exposition, which is a reworking of the previous article, provides a uniform analysis over all the families, including the previously undetermined ones. In the group \(C\rtimes \mathrm{Gal}(E/F)\), we use the Galois connection to calculate stabilizers of linear characters, and these stabilizers determine the full character degree set. This is shown for each subgroup \(C\leq E^\times\) which satisfies the condition that every prime dividing \(|E^\times :C|\) divides \(|F^\times|\).On double cosets with the trivial intersection property and Kazhdan-Lusztig cells in \(S_n\).https://www.zbmath.org/1456.200012021-04-16T16:22:00+00:00"McDonough, Thomas P."https://www.zbmath.org/authors/?q=ai:mcdonough.thomas-p"Pallikaros, Christos A."https://www.zbmath.org/authors/?q=ai:pallikaros.christos-aSummary: For a composition \(\lambda\) of \(n\) our aim is to obtain reduced forms for all the elements in the \(w_{J(\lambda)}\), the longest element of the standard parabolic subgroup of \(S_n\) corresponding to \(\lambda\). We investigate how far this is possible to achieve by looking at elements of the form \(w_{J(\lambda)}d\), where \(d\) is a prefix of an element of minimum length in a \((W_{J(\lambda)},B)\) double coset with the trivial intersection property, \(B\) being a parabolic subgroup of \(S_n\) whose type is ``dual'' to that of \(W_{J(\lambda)}\).On irreducibility of Koopman representations corresponding to measure contracting actions.https://www.zbmath.org/1456.200032021-04-16T16:22:00+00:00"Dudko, Artem"https://www.zbmath.org/authors/?q=ai:dudko.artemSummary: We introduce a notion of measure contracting actions and show that Koopman representations corresponding to ergodic measure contracting actions are irreducible. We also show that the actions of Higman-Thompson groups on intervals equipped with Lebesgue measure and the actions of weakly branch groups on the boundaries of rooted trees equipped with non-uniform Bernoulli measures are measure contracting. This gives a new point of view on irreducibility of the corresponding Koopman representations.