Recent zbMATH articles in MSC 20C30https://www.zbmath.org/atom/cc/20C302021-05-28T16:06:00+00:00WerkzeugRadicals of \(S_n\)-invariant positive semidefinite Hermitian forms.https://www.zbmath.org/1459.200062021-05-28T16:06:00+00:00"Franchi, Clara"https://www.zbmath.org/authors/?q=ai:franchi.clara"Ivanov, Alexander A."https://www.zbmath.org/authors/?q=ai:ivanov.alexander-a"Mainardis, Mario"https://www.zbmath.org/authors/?q=ai:mainardis.marioSummary: Let \(G\) be a finite group, \(V\) a complex permutation module for \(G\) over a finite \(G\)-set \(\mathcal{X}\), and \(f\colon V\times V \to \mathbb{C}\) a \(G\)-invariant positive semidefinite hermitian form on \(V\). In this paper we show how to compute the radical \(V^\bot\) of \(f\), by extending to nontransitive actions the classical combinatorial methods from the theory of association schemes. We apply this machinery to obtain a result for standard Majorana representations of the symmetric groups.Ordered set partitions, Garsia-Procesi modules, and rank varieties.https://www.zbmath.org/1459.053402021-05-28T16:06:00+00:00"Griffin, Sean T."https://www.zbmath.org/authors/?q=ai:griffin.sean-tSummary: We introduce a family of ideals \(I_{n,\lambda , s}\) in \(\mathbb{Q}[x_1,\dots , x_n]\) for \(\lambda\) a partition of \(k\leq n\) and an integer \(s \geq \ell (\lambda )\). This family contains both the Tanisaki ideals \(I_\lambda\) and the ideals \(I_{n,k}\) of \textit{J. Haglund} et al. [Adv. Math. 329, 851--915 (2018; Zbl 1384.05043)] as special cases. We study the corresponding quotient rings \(R_{n,\lambda , s}\) as symmetric group modules. When \(n=k\) and \(s\) is arbitrary, we recover the Garsia-Procesi modules, and when \(\lambda =(1^k)\) and \(s=k\), we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono [loc. cit.]. We give a monomial basis for \(R_{n,\lambda , s}\) in terms of \((n,\lambda , s)\)-staircases, unifying the monomial bases studied by \textit{A. M. Garsia} and \textit{C. Procesi} [ibid. 94, No. 1, 82--138 (1992; Zbl 0797.20012)] and Haglund-Rhoades-Shimozono [loc. cit.]. We realize the \(S_n\)-module structure of \(R_{n,\lambda , s}\) in terms of an action on \((n,\lambda , s)\)-ordered set partitions. We find a formula for the Hilbert series of \(R_{n,\lambda , s}\) in terms of inversion and diagonal inversion statistics on a set of fillings in bijection with \((n,\lambda , s)\)-ordered set partitions. Furthermore, we prove an expansion of the graded Frobenius characteristic of our rings into Gessel's fundamental quasisymmetric basis. We connect our work with Eisenbud-Saltman rank varieties using results of \textit{J. Weyman} [Invent. Math. 98, No. 2, 229--245 (1989; Zbl 0717.20033)]. As an application of our results on \(R_{n,\lambda , s}\), we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices.Cocharacters for the weak polynomial identities of the Lie algebra of \(3 \times 3\) skew-symmetric matrices.https://www.zbmath.org/1459.160232021-05-28T16:06:00+00:00"Domokos, Mátyás"https://www.zbmath.org/authors/?q=ai:domokos.matyas"Drensky, Vesselin"https://www.zbmath.org/authors/?q=ai:drensky.vesselinLet \(so_3\) be the Lie algebra of the skew-symmetric matrices of order 3 over a fixed field \(K\) of characteristic 0, and let \(M_3\) be the (associative) full matrix algebra of order 3 over \(K\). The authors study the weak polynomial identities for the pair \((M_3, so_3)\). If \(A\) is an associative algebra denote by \(A^-\) the Lie algebra on the vector space of \(A\) defined by the usual bracket \([a,b]=ab-ba\). Let us recall that if \(L\) is a Lie algebra and \(A\) an associative algebra such that \(L\subseteq A^-\) is a subalgebra and \(L\) generates \(A\), then an associative polynomial is a \textit{weak identity} for the pair \((A,L)\) whenever \(f\) vanishes on \(L\). Clearly the set \(I(A,L)\) of all weak identities for the pair \((A,L)\) is an ideal in the free associative algebra; this ideal is closed under Lie substitutions of the variables. Weak identities were introduced by \textit{J. P. Razmyslov} [Algebra Logic 12, 47--63 (1974; Zbl 0282.17003); translation from Algebra Logika 12, 83--113 (1973)], see also the monograph [\textit{Yu. P. Razmyslov}, Identities of algebras and their representations. Transl. from the Russian by A. M. Shtern. Transl. ed. by Simeon Ivanov. Providence, RI: American Mathematical Society (1994; Zbl 0827.17001)]. They were essential in establishing the finite basis property of the identities satisfied by \(sl_2\) and by \(M_2\). The weak identities were used by Razmyslov in order to construct central polynomials for the full matrix algebras as well. Weak identities have a wide field of applications in studying polynomial identities for non-associative algebras other than Lie algebras. Recall that, in a sense, the weak identities (in the Lie case) can be considered as identities of representations of Lie algebras. A basis (that is a generating set) of the ideal of weak identities for \((M_3, so_3)\) was described by Razmyslov, see the above cited monograph.
By using the isomorphism \(so_3\cong sl_2\) one can interpret the pair \((M_3, so_3)\) as \((M_3, ad(sl_2))\), where \(ad(sl_2)\) stands for the adjoint representation of \(sl_2\). When dealing with polynomial identities in characteristic 0 it is enough to consider the multilinear ones. If \(P_n\) is the vector space of all multilinear polynomials of degree \(n\) in the first \(n\) variables, in the free associative algebra then \(P_n\) is naturally a left module over the symmetric group \(S_n\); it acts by permuting the variables. If \(I\) is an ideal which is invariant under permutations from \(S_n\) then \(I\cap P_n\) is a submodule. Clearly ideals of (weak) identities are such ideals. When studying polynomial identities it is more convenient to work with the quotient \(P_n/(P_n\cap I)\). Its \(S_n\)-character is the \(n\)-th cocharacter of the ideal \(I\). Thus knowing the generators of \(I\) and its cocharacter sequence yields a very precise description of the identities in \(I\).
The paper under review studies the weak identities of the pair \((M_3, so_3)\). The authors determine explicitly the cocharacter of \(I(M_3, so_3)\) (see Lemma 3.3 and Theorem 3.7 of the paper). Although this is very significant contribution to the quantitative theory of polynomial identities the methods and results used to achieve it seem more important. The authors make use of the duality of the representations of \(S_n\) with the polynomial representations of the general linear group \(\mathrm{GL}_p\). Thus consider \(p\)-tuples of generic \(3\times 3\) skew-symmetric matrices \(t_1\), \dots, \(t_p\), acting on by the special orthogonal group \(\mathrm{SO}_3\) by simultaneous conjugation. Let \(T_p\) be the polynomial algebra in the \(3p\) variables that are the entries of the generic matrices \(t_i\), \(1\le i\le p\). Put \(\mathcal{F}_p\) the (associative and unitary) subalgebra of \(M_3(K[T_p])\) generated by the \(t_i\), and let \(\mathcal{E}_p\) be the subalgebra of \(M_3(K[T_p])\) consisting of the \(\mathrm{SO}_3(K)\)-equivariant polynomial maps from \(p\) copies of \(so_3\) to \(M_3\). The generators of \(\mathcal{E}_p\) are well known from classical invariant theory. The group \(\mathrm{GL}_p\) acts on the right on \(so_3^{\oplus p}\) in a natural way, and this action gives rise to one from the left on \(K[T_p]\) and on \(M_3(K[T_p])\). Clearly the actions of \(\mathrm{GL}_p\) and of \(\mathrm{SO}_3\) commute, hence \(\mathcal{E}_p\) and \(\mathcal{F}_p\) both become \(\mathrm{GL}_p\)-modules, \(\mathcal{F}_p\subset\mathcal{E}_p\). The authors obtain descriptions of the \(\mathrm{GL}_p\)-module structure of both \(\mathcal{F}_p\) and \(\mathcal{E}_p\), that is their decompositions into irreducibles, and compute the corresponding multiplicities. It should be noted that they compute the multiplicities in the case of \(\mathcal{E}_p\) by means of classical invariant theory. These clearly give upper bounds for the corresponding multiplicities of \(\mathcal{F}_p\). Afterwards by means of direct and heavy (CAS-aided) computations, they find the differences for \(\mathcal{F}_p\). In a sense, the study can be reduced to that of 3 generic matrices.
The paper will be of interest to people working on PI, on invariant theory, and representation theory of groups.
Reviewer: Plamen Koshlukov (Campinas)On decomposition numbers of diagram algebras.https://www.zbmath.org/1459.200082021-05-28T16:06:00+00:00"Shalile, Armin"https://www.zbmath.org/authors/?q=ai:shalile.arminSummary: In this paper, we survey an algorithm which determines the decomposition numbers of the partition algebra, Brauer algebra and walled Brauer algebra over a field of characteristic 0. The algorithm is based on the action of a set of distinguished elements of the algebra, the so-called Jucys-Murphy elements. We also outline the proof which is remarkably uniform.
For the entire collection see [Zbl 1394.14002].Some natural extensions of the parking space.https://www.zbmath.org/1459.053422021-05-28T16:06:00+00:00"Konvalinka, Matjaž"https://www.zbmath.org/authors/?q=ai:konvalinka.matjaz"Tewari, Vasu"https://www.zbmath.org/authors/?q=ai:tewari.vasu-vSummary: We construct a family of \(S_n\) modules indexed by \(c \in \{1, \ldots, n \}\) with the property that upon restriction to \(S_{n - 1}\) they recover the classical parking function representation of Haiman. The construction of these modules relies on an \(S_n\)-action on a set that is closely related to the set of parking functions. We compute the characters of these modules and use the resulting description to classify them up to isomorphism. In particular, we show that the number of isomorphism classes is equal to the number of divisors \(d\) of \(n\) satisfying \(d \neq 2\,\pmod 4\). In the cases \(c = n\) and \(c = 1\), we compute the number of orbits. Based on empirical evidence, we conjecture that when \(c = 1\), our representation is \(h\)-positive and is in fact the (ungraded) extension of the parking function representation constructed by \textit{A. Berget} and \textit{B. Rhoades} [J. Comb. Theory, Ser. A 123, 43--56 (2014; Zbl 1281.05130)].Representations of Brauer category and categorification.https://www.zbmath.org/1459.170442021-05-28T16:06:00+00:00"Rui, Hebing"https://www.zbmath.org/authors/?q=ai:rui.hebing"Song, Linliang"https://www.zbmath.org/authors/?q=ai:song.linliang
Let \(K\) be an algebraically closed field of characteristic zero or prime \(p \neq 2\). The authors consider the Brauer diagram category \({\mathcal B}(\delta_0)\) associated to an element \(\delta_0 \in K\) as introduced by \textit{G. Lehrer} and \textit{R. B. Zhang} [J. Eur. Math. Soc. 17, No. 9, 2311--2351 (2015; Zbl 1328.14079)]. The authors follow their previous work [Math. Z. 293, No. 1--2, 503--550 (2019; Zbl 07106142)] and consider \({\mathcal B}(\delta_0)\) as a subcategory of an affine Brauer categeory. Associated to \({\mathcal B}(\delta_0)\) is an infinite dimensional but locally unital and locally finite dimensional associative \(K\)-algebra \(\displaystyle B := \bigoplus_{m,n \in {\mathbb N}}\Hom_{{\mathcal B}(\delta_0)}(\mathsf{m},\mathsf{n})\), where \(\mathsf{m}\) is the object of the affine Brauer category associated to the integer \(m\). The algebra \(B\) is shown to be semisimple if and only if the characteristic is zero and \(\delta_0 \not\in {\mathbb Z}1_{K}\).
\(B\) contains a certain subalgebra \(B^0\) (defined diagrammatically) that may be identified as a direct sum of group algebras \(K{\mathfrak S}_m\) of symmetric groups over all \(m\). The authors define standardization and costandardization functors from the category of locally unital \(B^0\)-modules to locally unital \(B\)-modules, which restrict to maps on the subcategories of locally finite dimensional modules. Using Specht, Young, and simple modules for symmetric groups, the authors define standard and costandard modules associated to regular partitions. With these objects, it is shown that the category of locally finite dimensional \(B\)-modules is an upper finite fully stratified category in general and an upper finite highest weight category in the characteristic zero case, as defined in recent work of \textit{J. Brundan} and \textit{C. Stroppel} [``Semi-infinite highest weight categories,'' Preprint, \url{arXiv:180808022}].
The authors also focus on the category \(B\)-mod\(^{\Delta}\) of locally finite dimensional \(B\)-modules that admit a finite filtration whose sections are standard modules. Denote its Grothendieck group by \(K_0(B\text{-mod}^{\Delta})\). Let \(\mathfrak{sl}_K\) denote the complex Lie algebra \(\mathfrak{sl}_{\infty}\) when \(K\) has characteristic zero or \(\hat{\mathfrak{sl}}_p\) in positive characteristic. The authors construct an action of a Lie subalgebra \({\mathfrak g} \subset \mathfrak{sl}_K\) on \({\mathbb C}\otimes_{\mathbb Z} K_0(B\text{-mod}^{\Delta})\), where \({\mathfrak g}\) is all of \(\mathfrak{sl}_K\) when \(\delta_0 \not\in {\mathbb Z}1_{K}\) and is a certain proper subalgebra otherwise. As \({\mathfrak g}\)-modules, they show that \({\mathbb C}\otimes_{\mathbb Z} K_0(B\text{-mod}^{\Delta})\) is isomorphic to the (restriction of the) integrable highest weight \(\mathfrak{sl}_K\)-module with highest weight \(\varpi_d\) for \(d := \frac{\delta_0 - 1}{2}\). The proof makes use of an isomorphism between \(K_0(B\text{-mod}^{\Delta})\) and \(K_0(B\text{-pmod})\), the Grothendieck group of the subcategory of finitely generated projective \(B\)-modules. In the characteristic zero case, when \(\delta_0 \in {\mathbb Z}1_K\) (the non-semisimple case), this highest weight module may be identified with the \(d\)-sector \(\Lambda_d^{\infty}{\mathbb W}\) of the semi-infinite wedge space on the restricted dual \({\mathbb W}\) of the natural \(\mathfrak{sl}_{\infty}\)-module. Further, under this isomorphism, it is shown that the basis of standard modules for \({\mathbb C}\otimes_{\mathbb Z} K_0(B\text{-mod}^{\Delta})\) corresponds to the monomial basis for \(\Lambda_d^{\infty}{\mathbb W}\) and the basis of projective modules for \({\mathbb C}\otimes_{\mathbb Z} K_0(B\text{-mod}^{\Delta})\) corresponds to the quasi-canonical basis for \(\Lambda_d^{\infty}{\mathbb W}\).
Reviewer: Christopher P. Bendel (Menomonie)Varieties with at most cubic growth.https://www.zbmath.org/1459.170022021-05-28T16:06:00+00:00"Mishchenko, S."https://www.zbmath.org/authors/?q=ai:mishchenko.s-s|mishchenko.sergei-petrovich|mishchenko.s-e|mishchenko.s-g|mishchenko.s-v"Valenti, A."https://www.zbmath.org/authors/?q=ai:valenti.antonino|valenti.angelaLet \(\mathcal V\) be a variety of (also nonassociative) algebras over a field of characteristic 0. One of the most natural ways to measure the complexity of \(\mathcal V\) is in terms of the asymptotic behavior of the codimension sequence \(c_n({\mathcal V})\), \(n=1,2,\ldots\). Due to the already classical results in [\textit{A. Regev}, Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007); \textit{A. Giambruno} and \textit{M. Zaicev}, Adv. Math. 140, No. 2, 145--155 (1998; Zbl 0920.16012); ibid. 142, No. 2, 221--243 (1999; Zbl 0920.16013)] the codimension sequence behaves nicely for varieties of associative algebras: \(c_n({\mathcal V})\) is exponentially bounded, \(\exp({\mathcal V})=\lim_{n\to\infty}\sqrt[n]{c_n({\mathcal V})}\) exists and is a nonnegative integer. When \(c_n({\mathcal V})\) is polynomially bounded the reviewer showed [\textit{V. Drensky}, Contemp. Math. 131, 285--300 (1992; Zbl 0766.16012)] that the growth of \(c_n({\mathcal V})\) is exactly polynomial. The situation is much worse in the nonassociative case. The codimension sequence may have an overexponential growth [\textit{V. M. Petrogradskiĭ}, Sb. Math. 188, No. 6, 913--931 (1997; Zbl 0890.17002); translation from Mat. Sb. 188, No. 6, 119--138 (1997)] and in the case of the polynomial growth the codimension sequence may have a fractional polynomial growth [\textit{M. V. Zaĭtsev} and \textit{S. P. Mishchenko}, Mosc. Univ. Math. Bull. 63, No. 1, 25--31 (2008; Zbl 1199.17001); translation from Vestn. Mosk. Univ., Ser. I 2008, No. 1, 25--31 (2008)].
In the paper under review the authors study varieties \(\mathcal V\) of nonassociative algebras satisfying the polynomial identity \(x(yz)=0\) such that \(c_n({\mathcal V})\leq Cn^a\), where \(C>0\) and \(a<3\). They show that if \(a<2\) then the growth is linear, i.e. \(c_n({\mathcal V})\leq C_1n\) and when \(2\leq a<3\) then the growth is quadratic, i.e. \(c_n({\mathcal V})\leq C_2n^2\) for some positive constants \(C_1, C_2\). This is the best possible result in this direction because \(c_n({\mathcal V})={\mathcal O}(n^{7/2})\) for the variety \(\mathcal V\) in the example of Zaĭtsev and Mishchenko cited above.
Reviewer: Vesselin Drensky (Sofia)On the symmetric and exterior powers of Young permutation modules.https://www.zbmath.org/1459.200072021-05-28T16:06:00+00:00"Jiang, Yu"https://www.zbmath.org/authors/?q=ai:jiang.yu.3|jiang.yu|jiang.yu.4|jiang.yu.1|jiang.yu.2Let \(G\) be a finite group and \(\mathbb{F}\) be a field of finite characteristic \(p\). If \(M\) and \(N\) are \(\mathbb{F}G\) modules, then we write \(M\mid N\) if \(M\) is a isomorphic to a direct summand of \(N\); and \(S^{a}M\) and \(\Lambda^{a}M\) to denote the \(a\)-th symmetric power (respectively, exterior power) of \(M\) for \(a=1,2,\dots\). Let \(c_{G}(M)\) denote the length of the shortest projective resolution of \(M\). The present paper considers the problem of finding \(c_{G}(S^{a}G)\) and \(c_{G}(\Lambda^{a}G)\) for finite groups, particularly when
\(G\) is a symmetric group \(\mathfrak{S}_{n}\). Some these results strengthen earlier known results. The following are some examples.
Theorem A: If \(G\) is a finite group of \(p\)-rank \(r\), and \(P\neq0\) is a finitely generated projective \(\mathbb{F}G\)-module, then for each \(a\) we have \(c_{G}(S^{a}P)=\min(\nu_{p}(a),r)\) where \(p^{\nu_{p}(a)}\parallel a\); moreover, \(c_{G}(\Lambda^{a}P)=c_{G}(S^{a}P)\) whenever \(a\leq\dim_{\mathbb{F}}P\).
Assume \(G=\mathfrak{S}_{n}\), let \(M^{\lambda}\) be the \(\mathbb{F}\mathfrak{S}_{n}\)-Young permutation module for a partition \(\lambda\vdash n\), and let \(Y^{\lambda}\) be the corresponding (indecomposable) Specht module (so \(Y^{\lambda}\) is a direct summand of multiplicity \(1\) in \(M^{\lambda}\)).
Theorem B: For each \(a>1\) and \(\lambda\vdash n\) there exists some \(\mu\vdash n\) such that \(Y^{\mu}\mid S^{a}M^{\lambda}\) and \(c_{\mathfrak{S}_{n}}(Y^{\mu})=c_{\mathfrak{S}_{n}}(S^{a}M^{\lambda})\); moreover, if \(\lambda\neq(n)\) then there exists \(\mu\) such that \(Y^{\mu}\mid\Lambda ^{2}M^{\lambda}\) and \(c_{\mathfrak{S}_{n}}(Y^{\mu})=c_{\mathfrak{S}_{n}}(\Lambda^{a}M^{\lambda})\).
Corollary 3.7: If \(\lambda=(\lambda_{1},...,\lambda_{\ell})\) then \(S^{a}M^{\lambda}\) is projective if and only if either (i) \(n < p\) or (ii) \(p\nmid a\) and \(\lambda_{i} < p\) for all \(i\).
Corresponding, but more complicated, theorems describe the projective modules of the form \(\Lambda^{a}M^{\lambda}\) and the value of \(c_{\mathfrak{S}_{n}}(\Lambda^{2}M^{\lambda})\).
Reviewer: John D. Dixon (Ottawa)