scholarly journals $q,t$-Catalan Numbers and Generators for the Radical Ideal defining the Diagonal Locus of $({\mathbb C}^2)^n$

10.37236/645 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li
Keyword(s):  
Vector Space ◽  
Hilbert Series ◽  
Upper Bounds ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Minimal Set ◽  
Radical Ideal ◽  
The Ideal

Let $I$ be the ideal generated by alternating polynomials in two sets of $n$ variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the bi-graded vector space $M(=\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by a minimal set of generators for $I$. In this paper we give simple upper bounds on $\text{dim }M_{d_1, d_2}$ in terms of number of partitions, and find all bi-degrees $(d_1,d_2)$ such that $\dim M_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $M_{d_1, d_2}$.

scholarly journals On the diagonal ideal of $(\mathbb{C}^2)^n$ and $q,t$-Catalan numbers

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li
Keyword(s):  
Vector Space ◽  
Hilbert Series ◽  
Upper Bounds ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Nous Obtenons ◽  
Minimal Set ◽  
Nous Trouvons ◽  
International Audience

International audience Let $I_n$ be the (big) diagonal ideal of $(\mathbb{C}^2)^n$. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M_n=\bigoplus_{d_1,d_2}(M_n)_{d_1,d_2}$ spanned by a minimal set of generators for $I_n$. We give simple upper bounds on $\textrm{dim} (M_n)_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\textrm{dim} (M_n)_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $(M_n)_{d_1, d_2}$. Soit $I_n$ l'idéal de la (grande) diagonale de $(\mathbb{C}^2)^n$. Haiman a démontré que le $q,t$-nombre de Catalan est la série de Hilbert de l'espace vectoriel gradué $M_n=\bigoplus_{d_1,d_2}(M_n)_{d_1,d_2}$ engendré par un ensemble minimal de générateurs de $I_n$. Nous obtenons des bornes supérieures simples pour $\textrm{dim} (M_n)_{d_1, d_2}$ en termes de nombres de partitions, ainsi que tous les bi-degrés $(d_1, d_2)$ pour lesquels ces bornes supérieures sont atteintes. Pour ces bi-degrés, nous trouvons aussi des bases explicites de $(M_n)_{d_1, d_2}$.

Robinson–Schensted–Knuth Correspondence and Weak Polynomial Identities of M1,1(E)

2005 ◽  
Vol 12 (02) ◽  
pp. 333-349
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Roberto La Scala
Keyword(s):  
Hilbert Series ◽  
Quotient Algebra ◽  
Infinite Field ◽  
Polynomial Identities ◽  
The Ideal

In this paper, it is proved that the ideal Iw of the weak polynomial identities of the superalgebra M1,1(E) is generated by the proper polynomials [x1, x2, x3] and [x2, x1] [x3, x1] [x4, x1]. This is proved for any infinite field F of characteristic different from 2. Precisely, if B is the subalgebra of the proper polynomials of F<X>, we determine a basis and the dimension of any multihomogeneous component of the quotient algebra B / (B ∩ Iw). We also compute the Hilbert series of this algebra. One of the main tools of this paper is a variant we found of the Robinson–Schensted–Knuth correspondence defined for single semistandard tableaux of double shape.

scholarly journals Amount algebras

2021 ◽  
pp. 2250102
Author(s):  
Peyman Nasehpour
Keyword(s):  
Positive Integer ◽  
Valuation Ring ◽  
Prüfer Domain ◽  
Radical Ideal ◽  
Ideal Formula ◽  
Torsion Free ◽  
The Ideal

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson–Badawi [Formula: see text] conjecture, we prove that under some conditions, the formula [Formula: see text] holds for some amount [Formula: see text]-algebras [Formula: see text] and some ideals [Formula: see text] of [Formula: see text], where [Formula: see text] is the smallest positive integer [Formula: see text] that the ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing. A corollary to the mentioned formula is that if, for example, [Formula: see text] is a Prüfer domain or a torsion-free valuation ring and [Formula: see text] is a radical ideal of [Formula: see text], then [Formula: see text].

scholarly journals On the dimension of the "cohits" space $\mathbb Z_2\otimes_{\mathcal A_2} H^{*}((\mathbb RP(\infty))^{\times t}, \mathbb Z_2)$ and some applications

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Negative Integer ◽  
Central Problem ◽  
Minimal Set ◽  
Prime Field ◽  
Monomial Basis ◽  
Hit Problem

We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with the degree of each $x_i$ being one. Let $\mathcal A_2$ be the Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the $\mathbb Z_2$-graded vector space $\mathbb Z_2\otimes_{\mathcal A_2} P_t.$ This problem, which is called the "hit" problem for Steenrod algebra, has been systematically studied for $t\leq 4.$ The present paper is devoted to the investigation of the structure of the "cohits" space $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ in some certain "generic" degrees. More specifically, we explicitly determine a monomial basis of $\mathbb Z_2\otimes_{\mathcal A_2} P_5$ in degree \mbox{$n_s=5(2^{s}-1) + 42.2^{s}$} for every non-negative integer $s.$ As a result, it confirms Sum's conjecture \cite{N.S2} for a relation between the minimal sets of $\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and degree $n_s$. Based on Kameko's map \cite{M.K} and a previous result by Sum \cite{N.S1}, we obtain a inductive formula for the dimension of $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ in a generic degree given. As an application, we obtain the dimension of $\mathbb Z_2\otimes_{\mathcal A_2} P_6$ in the generic degree $5(2^{s+5}-1) + n_0.2^{s+5}$ for all $s\geq 0,$ and show that the Singer's cohomological transfer \cite{W.S1} is an isomorphism in bidegree $(5, 5+n_s)$.

scholarly journals The ideal structure of idempotent-generated transformation semigroups

1985 ◽  
Vol 28 (3) ◽  
pp. 319-331 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Boolean Ring ◽  
Ideal Structure ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Ideal

Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.

On the reduction numbers of monomial ideals

2019 ◽  
Vol 19 (10) ◽  
pp. 2050201
Author(s):  
Ibrahim Al-Ayyoub
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Upper Bounds ◽  
Integral Closure ◽  
Monomial Ideals ◽  
Explicit Method ◽  
Generating Set ◽  
Ideal Formula ◽  
Reduction Number ◽  
The Ideal

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

scholarly journals Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application to Singer's cohomological transfer

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Open Problem ◽  
General Linear Group ◽  
Homotopy Theory ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Ext Groups ◽  
Hit Problem ◽  
Prime 2

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This problem is the content of "hit problem" of Frank Peterson. We study the $q$-th Singer algebraic transfer $Tr_q^{A}$, which is a homomorphism from the space of $GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the Adams $E_2$-term, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The Singer transfer is one of the useful tools for describing mysterious Ext groups. In the present study, by using techniques of the hit problem of four variables, we explicitly determine the structure of the spaces $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and the representation of the fourth transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. These new results confirmed Singer's conjecture for the monomorphism of the rank $4$ transfer. Our approach is different from that of Singer.

Minimal Free Resolutions of Zero-dimensional Schemes in ℙ1 × ℙ1

2015 ◽  
Vol 22 (01) ◽  
pp. 97-108 ◽  
Author(s):  
Paola Bonacini ◽  
Lucia Marino
Keyword(s):  
Free Resolution ◽  
Hilbert Functions ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Minimal Set ◽  
Minimal Free Resolutions ◽  
The Ideal

Let X be a zero-dimensional scheme in ℙ1 × ℙ1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depend only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines.

COMMENTS REGARDING d-IDEALS OF CERTAIN f-RINGS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350008 ◽  
Author(s):  
THEMBA DUBE ◽  
OGHENETEGA IGHEDO
Keyword(s):  
Radical Ideal ◽  
Baer Ring ◽  
Ring Of Fractions ◽  
F Ring ◽  
The Ideal

Let A be a reduced commutative f-ring with identity and bounded inversion, and let A* be its subring of bounded elements. By first observing that A is the ring of fractions of A* relative to the subset of A* consisting of elements which are units in the bigger ring, we show that the frames Did (A) and Did (A*) of d-ideals of A and A*, respectively, are isomorphic, and that the isomorphism witnessing this is precisely the restriction of the extension map I ↦ Ie which takes a radical ideal of A* to the ideal it generates in A. Specializing to the ring [Formula: see text], we show that if L is an F-frame, then the saturation quotient of [Formula: see text] is isomorphic to βL. We also investigate projectability properties of [Formula: see text] and [Formula: see text], where the latter denotes the frame of z-ideals of [Formula: see text]. We show that [Formula: see text] is flatly projectable precisely when [Formula: see text] is a feebly Baer ring. Quite easily, [Formula: see text] is projectable if and only if L is basically disconnected. Less obvious is that [Formula: see text] is projectable if and only if L is cozero-complemented.

scholarly journals MODIFIKASI ALGORITMA VIGENÈRE CIPHER MENGGUNAKAN METODE CATALAN NUMBER DAN DOUBLE COLUMNAR TRANSPOSITION

Compiler ◽  
2015 ◽  
Vol 4 (1) ◽  
Author(s):  
Guruh Marindra Pratama ◽  
E.Nurmiyati Tamatjita
Keyword(s):  
Mathematical Method ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Slowing Down ◽  
The Past

Vigenère Cipher is one of the well-known ciphering algorithms of the past. Modifications to Vigenère Cipher algorithm is made to improve its security, making it more difficult to decipher by a cryptanalyst. Due to the nature of the algorithm, these measures have to be taken to maintain the confidentiality of ciphered data.  This research modified the Vigenère Cipher using Catalan Numbers method and Double Columnar Transposition.  Catalan Numbers method is a mathematical method used to randomize the initial key so as to generate a key which is longer and having stronger characteristics; a key which is harder to guess, either by cryptanalysts or by key-deciphering methods. In addition to the first method, Double Columnar Transposition is used to rearrange the position of data in the generated ciphertext in order to make it appear more random, hence slowing down the cryptanalysis process of the encrypted text. Double Columnar Transposition is done by applying columnar transposition twice to the ciphered text. The applied modifications to Vigenère Cipher are then tested using Kasiski Examination. Resulting ciphertexts are known to have randomised characteristics, which made it difficult to guess the ciphering method used to generate the ciphertexts. Tests done using Kasiski Examination {1, 2, 4} proven that the ciphertexts passed the test, hence putting down the possibility of easy deciphering, and the modifications successfully provided a better and stronger encryption to the Vigenère Cipher.

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