The intent of this paper is to give the reader, in a general sense, how to go about finding irreducible representations of the Symmetric Group Sn. While I would like to be thorough toward this end, I fear we must assume some results from Wedderburn Theory that will be given without proof because although they are important, proving and discussing these results is not in the scope of this paper....

Pfaffians of matrices with entries z[i, j]/(xi + xj), or determinants of matrices with entries z[i, j]/(xi−xj), where the antisymmetrical indeterminates z[i, j] satisfy the Plücker relations, can be identified with a trace in an irreducible representation of a product of two symmetric groups. Using Young’s orthogonal bases, one can write explicit expressions of such Pfaffians and determinants, ...

Remark 1.2. The definition of an FG-module is more technical than the definition of a representation of G, but, as the exercise shows, the two notions are equivalent. Module can be more convenient to work with, because there is less notation, and we can use results from ring theory without any translation. The language of representations is preferable if we want to have an explicit map ρ : G→ G...

In [2], Dixon considered the following question: " Suppose two permutations are chosen at random from the symmetric group Sn of degree n. What is the probability that they will generate Sn? " Actually, Netto conjectured last century that almost all pairs of elements from Sn will generate Sn or An. Dixon showed that this is true in the following sense: The proportion of ordered pairs (x, y) (x, ...

In regard to the linear subspace T (n) of n n symmetric Toeplitz matrices over the real eld, the collection S(n) of all real and orthogonal matrices Q such that QTQT 2 T (n) whenever T 2 T (n) forms a group, called the stability group of T (n). This paper shows that S(n) is nite. In fact, S(n) has exactly eight elements regardless of the dimension n. Group elements in S(n) are completely charac...

where χ(1) denotes the dimension of the character χ and (n)k = n(n − 1) · · · (n − k + 1). Thus [8, (7.6)(ii)][12, p. 349] χ(1) is the number f of standard Young tableaux of shape λ. Identify λ with its diagram {(i, j) : 1 ≤ j ≤ λi}, and regard the points (i, j) ∈ λ as squares (forming the Young diagram of λ). We write diagrams in “English notation,” with the first coordinate increasing from to...

Group signatures are used extensively for privacy in anonymous credentials schemes and in real-world systems for hardware enclave attestation. As such, there is a strong interest in making these schemes post-quantum secure. In this paper we initiate the study of group signature schemes built only from symmetric primitives, such as hash functions and PRFs, widely regarded as the safest primitive...

si def = jfj; xj = igj. Let s be a composition, that is to say a q-tuple which is the composition of some x in X; we set Xs = fx 2 X; s(x) = sg. We de ne and study a notion of designs for subsets of F q . In the case q = 2, our de nition coincides with the usual notion of designs and in fact our de nition is equivalent to the notion of strong colored designs of A. Bonnecaze, P. Sol e, P. Udaya ...

Given a natural number n ∈ N, a partition λ ` n is a decomposition of n into an increasing sum of natural numbers. For example, associated to the sum 3+2+2+1 = 8 we have the partiton λ = (3, 2, 2, 1) ` 8. One way mathematicians study partitions is via Young diagrams which are collections of n top and left justified boxes with rows corresponding to the elements of λ. Figure 1 shows an example of...

In this paper, we determine all of subgroups of symmetric group S4 by applying Lagrange theorem and Sylow theorem. First, we observe the multiplication table of S4, then we determine all possibilities of every subgroup of order n, with n is the factor of order S4. We found 30 subgroups of S4. The diagram of lattice subgroups of S4 is then presented. Mathematics Subject Classification: 20B30, 20...