Skew characters and cyclic sieving

In 2010, Rhoades proved that promotion on rectangular standard Young tableaux, together with the associated fake-degree polynomial, provides an instance of the cyclic sieving phenomenon. We extend this result to m-tuples of skew standard Young tableaux of the same shape, for fixed m, subject to the condition that the mth power of the associated fake-degree polynomial evaluates to nonnegative integers at roots of unity. However, we are unable to specify an explicit group action. Put differently, we determine in which cases the mth tensor power of a skew character of the symmetric group carries a permutation representation of the cyclic group. To do so, we use a method proposed by Amini and the first author, which amounts to establishing a bound on the number of border-strip tableaux of skew shape. Finally, we apply our results to the invariant theory of tensor powers of the adjoint representation of the general linear group. In particular, we prove the existence of a bijection between permutations and Stembridge’s alternating tableaux, which intertwines rotation and promotion.

Immutable and Privacy Protected E-Certificate Repository on Blockchain

Fake education certificates or fake degree is one of the major concerns in higher education. This fraud can be minimized if there is a tamper-proof and confidential registry of certificates wherein not one but multiple certified authorities verifies and stores the issued certificate in immutable repositories with proper privacy maintained. Secondly, there should be a mechanism for retrieving the authentic certificate without much cost and time. Blockchain is an immutable, shared, distributed ledger without the control of a single centralized authority that fits very well for the discussed use case. The proposed work, PrivateCertChain, has implemented the idea for university having multiple affiliated colleges, by deploying and verifying digitally signed e-certificate on Ethereum Blockchain. Multiple affiliated colleges can serve as the miners for verifying the signature of the issuer. For privacy concerns, the content of the certificate will be hashed and this hashed value will be stored in Blockchain along with the roll number of the certificate holder. Once the transaction hash is generated, it will be converted to QR code. The QR code is shared with the respective owner of the certificate and it will also serve as the credential of the certificate. Thus, anyone having the credential can view the authentic certificate which is kept on the blockchain, by scanning QR through the dedicated application designed for verification. The proposed solution can be a foolproof mechanism against all frauds as it guards for integrity, confidentiality, authenticity, and privacy of educational certificates.

Invariant Tensors and the Cyclic Sieving Phenomenon

We construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\mathfrak{S}_r$ related to the natural action of $\mathfrak{S}_r$ on the subspace of invariant tensors in $\otimes^rM$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux.

Units of group rings, the Bogomolov multiplier and the fake degree conjecture

Let π be a finite p-group and ${\mathbb{F}_{q}}$ a finite field with q = pn elements. Denote by $\I_{\mathbb{F}_{q}}$ the augmentation ideal of the group ring ${\mathbb{F}_{q}}$[π]. We have found a surprising relation between the abelianization of 1 + $\I_{\mathbb{F}_{q}}$, the Bogomolov multiplier B0(π) of π and the number of conjugacy classes k(π) of π: $$ \left | (1+\I_{\Fq})_{\ab} \right |=q^{\kk(\pi)-1}|\!\B_0(\pi)|. In particular, if π is a finite p-group with a non-trivial Bogomolov multiplier, then 1 + $\I_{\mathbb{F}_{q}}$ is a counterexample to the fake degree conjecture proposed by M. Isaacs.